Topological vector space

<h2 id="motivation">Motivation</h2>
<h3>Normed spaces</h3>
Every <a href="/facts/Normed_vector_space/rmDljfyE">normed vector space</a> has a natural <a href="/facts/Normed_vector_space/rmDljfyE">topological structure</a>: the norm induces a <a href="/facts/Metric_space/gnBBRXtc">metric</a> and the metric induces a topology.
This is a topological vector space because:

<ol><li>The vector addition map 
 
 
 
 ⋅
 
 +
 
 ⋅
 
 :
 X
 ×
 X
 →
 X
 
 
 {\displaystyle \cdot \,+\,\cdot \;:X\times X\to X}
 
 defined by 
 
 
 
 (
 x
 ,
 y
 )
 ↦
 x
 +
 y
 
 
 {\displaystyle (x,y)\mapsto x+y}
 
 is (jointly) continuous with respect to this topology. This follows directly from the <a href="/facts/Triangle_inequality/KubLjkwr">triangle inequality</a> obeyed by the norm.</li>
<li>The scalar multiplication map 
 
 
 
 ⋅
 :
 
 K
 
 ×
 X
 →
 X
 
 
 {\displaystyle \cdot :\mathbb {K} \times X\to X}
 
 defined by 
 
 
 
 (
 s
 ,
 x
 )
 ↦
 s
 ⋅
 x
 ,
 
 
 {\displaystyle (s,x)\mapsto s\cdot x,}
 
 where 
 
 
 
 
 K
 
 
 
 {\displaystyle \mathbb {K} }
 
 is the underlying scalar field of 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 is (jointly) continuous. This follows from the triangle inequality and homogeneity of the norm.</li></ol>
Thus all <a href="/facts/Banach_space/sDEIk7EC">Banach spaces</a> and <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert spaces</a> are examples of topological vector spaces.

<h3>Non-normed spaces</h3>
There are topological vector spaces whose topology is not induced by a norm, but are still of interest in analysis. Examples of such spaces are spaces of <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic functions</a> on an open domain, spaces of <a href="/facts/Infinitely_differentiable_function/mffL4ch2">infinitely differentiable functions</a>, the <a href="/facts/Schwartz_space/B035faw7">Schwartz spaces</a>, and spaces of <a href="/facts/Test_function/yfaJ4mUp">test functions</a> and the spaces of <a href="/facts/Distribution_(mathematics)/yfaJ4mUp">distributions</a> on them.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a> These are all examples of <a href="/facts/Montel_space/HLXqXHzs">Montel spaces</a>. An infinite-dimensional Montel space is never normable. The existence of a norm for a given topological vector space is characterized by <a href="/facts/Kolmogorov%2527s_normability_criterion/ckaPRrE9">Kolmogorov's normability criterion</a>. 
A <a href="/facts/Topological_field/qxQqzhCy">topological field</a> is a topological vector space over each of its <a href="/facts/Field_extension/L7yIDtyK">subfields</a>.

<h2 id="definition">Definition</h2>

A topological vector space (TVS) 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> over a <a href="/facts/Topological_field/qxQqzhCy">topological field</a> 
 
 
 
 
 K
 
 
 
 {\displaystyle \mathbb {K} }
 
 (most often the <a href="/facts/Real_number/R02gw5Pb">real</a> or <a href="/facts/Complex_number/2w7mqI7e">complex</a> numbers with their standard topologies) that is endowed with a <a href="/facts/Topological_space/v2ut7CbK">topology</a> such that vector addition 
 
 
 
 ⋅
 
 +
 
 ⋅
 
 :
 X
 ×
 X
 →
 X
 
 
 {\displaystyle \cdot \,+\,\cdot \;:X\times X\to X}
 
 and scalar multiplication 
 
 
 
 ⋅
 :
 
 K
 
 ×
 X
 →
 X
 
 
 {\displaystyle \cdot :\mathbb {K} \times X\to X}
 
 are <a href="/facts/Continuous_function_(topology)/sKbl02pB">continuous functions</a> (where the domains of these functions are endowed with <a href="/facts/Product_topology/0hg02IaX">product topologies</a>). Such a topology is called a vector topology or a TVS topology on 
 
 
 
 X
 .
 
 
 {\displaystyle X.}

Every topological vector space is also a commutative <a href="/facts/Topological_group/cAgNMMRU">topological group</a> under addition.
Hausdorff assumption
Many authors (for example, <a href="/facts/Walter_Rudin/9mxRAECY">Walter Rudin</a>), but not this page, require the topology on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 to be <a href="/facts/T1_space/K4exfuf7">T1</a>; it then follows that the space is <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff</a>, and even <a href="/facts/Tychonoff_space/jyCgpKB5">Tychonoff</a>. A topological vector space is said to be separated if it is Hausdorff; importantly, "separated" does not mean <a href="/facts/Separable_space/2bmU73bk">separable</a>. The topological and linear algebraic structures can be tied together even more closely with additional assumptions, the most common of which are listed below.
Category and morphisms
The <a href="/facts/Category_(category_theory)/7WzxUThf">category</a> of topological vector spaces over a given topological field 
 
 
 
 
 K
 
 
 
 {\displaystyle \mathbb {K} }
 
 is commonly denoted 
 
 
 
 
 
 T
 V
 S
 
 
 
 K
 
 
 
 
 
 {\displaystyle \mathrm {TVS} _{\mathbb {K} }}
 
 or 
 
 
 
 
 
 T
 V
 e
 c
 t
 
 
 
 K
 
 
 
 .
 
 
 {\displaystyle \mathrm {TVect} _{\mathbb {K} }.}
 
 The <a href="/facts/Object_(category_theory)/7WzxUThf">objects</a> are the topological vector spaces over 
 
 
 
 
 K
 
 
 
 {\displaystyle \mathbb {K} }
 
 and the <a href="/facts/Morphism/Kdpuyz3S">morphisms</a> are the <a href="/facts/Continuous_linear_map/KcqjDBrJ">continuous 
 
 
 
 
 K
 
 
 
 {\displaystyle \mathbb {K} }
 
-linear maps</a> from one object to another.
A topological vector space homomorphism (abbreviated TVS homomorphism), also called a <a href="/facts/Topological_homomorphism/rnnTWdNf">topological homomorphism</a>,<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a><a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> is a <a href="/facts/Continuous_map/sKbl02pB">continuous</a> <a href="/facts/Linear_map/Q1PBKNVV">linear map</a> 
 
 
 
 u
 :
 X
 →
 Y
 
 
 {\displaystyle u:X\to Y}
 
 between topological vector spaces (TVSs) such that the induced map 
 
 
 
 u
 :
 X
 →
 Im
 ⁡
 u
 
 
 {\displaystyle u:X\to \operatorname {Im} u}
 
 is an <a href="/facts/Open_mapping/abHUKX7l">open mapping</a> when 
 
 
 
 Im
 ⁡
 u
 :=
 u
 (
 X
 )
 ,
 
 
 {\displaystyle \operatorname {Im} u:=u(X),}
 
 which is the range or image of 
 
 
 
 u
 ,
 
 
 {\displaystyle u,}
 
 is given the <a href="/facts/Subspace_topology/NjU6z4Ca">subspace topology</a> induced by 
 
 
 
 Y
 .
 
 
 {\displaystyle Y.}

A topological vector space embedding (abbreviated TVS embedding), also called a topological <a href="/facts/Monomorphism/Q9BrtDTs">monomorphism</a>, is an <a href="/facts/Injective_map/5ES6tqf5">injective</a> topological homomorphism. Equivalently, a TVS-embedding is a linear map that is also a <a href="/facts/Topological_embedding/HKeVKc42">topological embedding</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>
A topological vector space isomorphism (abbreviated TVS isomorphism), also called a topological vector isomorphism<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> or an isomorphism in the category of TVSs, is a bijective <a href="/facts/Linear_map/Q1PBKNVV">linear</a> <a href="/facts/Homeomorphism/jMfWg3Mn">homeomorphism</a>. Equivalently, it is a <a href="/facts/Surjective_map/KezhLpCb">surjective</a> TVS embedding<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>
Many properties of TVSs that are studied, such as <a href="/facts/Locally_convex_topological_vector_space/AGrPrayC">local convexity</a>, <a href="/facts/Metrizable_topological_vector_space/6FUHfx6W">metrizability</a>, <a href="/facts/Complete_topological_vector_space/e6qtS76j">completeness</a>, and <a href="/facts/Normable_space/rmDljfyE">normability</a>, are invariant under TVS isomorphisms.
A necessary condition for a vector topology
A collection 
 
 
 
 
 
 N
 
 
 
 
 {\displaystyle {\mathcal {N}}}
 
 of subsets of a vector space is called additive<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> if for every 
 
 
 
 N
 ∈
 
 
 N
 
 
 ,
 
 
 {\displaystyle N\in {\mathcal {N}},}
 
 there exists some 
 
 
 
 U
 ∈
 
 
 N
 
 
 
 
 {\displaystyle U\in {\mathcal {N}}}
 
 such that 
 
 
 
 U
 +
 U
 ⊆
 N
 .
 
 
 {\displaystyle U+U\subseteq N.}

Characterization of continuity of addition at 
 
 
 
 0
 
 
 {\displaystyle 0}
 
<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>—If 
 
 
 
 (
 X
 ,
 +
 )
 
 
 {\displaystyle (X,+)}
 
 is a <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> (as all vector spaces are), 
 
 
 
 τ
 
 
 {\displaystyle \tau }
 
 is a topology on 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 and 
 
 
 
 X
 ×
 X
 
 
 {\displaystyle X\times X}
 
 is endowed with the <a href="/facts/Product_topology/0hg02IaX">product topology</a>, then the addition map 
 
 
 
 X
 ×
 X
 →
 X
 
 
 {\displaystyle X\times X\to X}
 
 (defined by 
 
 
 
 (
 x
 ,
 y
 )
 ↦
 x
 +
 y
 
 
 {\displaystyle (x,y)\mapsto x+y}
 
) is continuous at the origin of 
 
 
 
 X
 ×
 X
 
 
 {\displaystyle X\times X}
 
 if and only if the set of <a href="/facts/Neighborhood_(topology)/XHKgrpkp">neighborhoods</a> of the origin in 
 
 
 
 (
 X
 ,
 τ
 )
 
 
 {\displaystyle (X,\tau )}
 
 is additive. This statement remains true if the word "neighborhood" is replaced by "open neighborhood."

All of the above conditions are consequently a necessity for a topology to form a vector topology.

<h3>Defining topologies using neighborhoods of the origin</h3>
Since every vector topology is translation invariant (which means that for all 
 
 
 
 
 x
 
 0
 
 
 ∈
 X
 ,
 
 
 {\displaystyle x_{0}\in X,}
 
 the map 
 
 
 
 X
 →
 X
 
 
 {\displaystyle X\to X}
 
 defined by 
 
 
 
 x
 ↦
 
 x
 
 0
 
 
 +
 x
 
 
 {\displaystyle x\mapsto x_{0}+x}
 
 is a <a href="/facts/Homeomorphism/jMfWg3Mn">homeomorphism</a>), to define a vector topology it suffices to define a <a href="/facts/Neighborhood_basis/fMRM8Xht">neighborhood basis</a> (or subbasis) for it at the origin.

Theorem<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> (Neighborhood filter of the origin)—Suppose that 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a real or complex vector space. If 
 
 
 
 
 
 B
 
 
 
 
 {\displaystyle {\mathcal {B}}}
 
 is a <a href="/facts/Empty_set/ffEk3eA6">non-empty</a> additive collection of <a href="/facts/Balanced_set/Y2ar7E6g">balanced</a> and <a href="/facts/Absorbing_set/GCtjsUuX">absorbing</a> subsets of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 then 
 
 
 
 
 
 B
 
 
 
 
 {\displaystyle {\mathcal {B}}}
 
 is a <a href="/facts/Neighborhood_base/fMRM8Xht">neighborhood base</a> at 
 
 
 
 0
 
 
 {\displaystyle 0}
 
 for a vector topology on 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 
 That is, the assumptions are that 
 
 
 
 
 
 B
 
 
 
 
 {\displaystyle {\mathcal {B}}}
 
 is a <a href="/facts/Filter_base/7f3zk55u">filter base</a> that satisfies the following conditions:

<ol><li>Every 
 
 
 
 B
 ∈
 
 
 B
 
 
 
 
 {\displaystyle B\in {\mathcal {B}}}
 
 is <a href="/facts/Balanced_set/Y2ar7E6g">balanced</a> and <a href="/facts/Absorbing_set/GCtjsUuX">absorbing</a>,</li>
<li>
 
 
 
 
 
 B
 
 
 
 
 {\displaystyle {\mathcal {B}}}
 
 is additive: For every 
 
 
 
 B
 ∈
 
 
 B
 
 
 
 
 {\displaystyle B\in {\mathcal {B}}}
 
 there exists a 
 
 
 
 U
 ∈
 
 
 B
 
 
 
 
 {\displaystyle U\in {\mathcal {B}}}
 
 such that 
 
 
 
 U
 +
 U
 ⊆
 B
 ,
 
 
 {\displaystyle U+U\subseteq B,}
 
</li></ol>
If 
 
 
 
 
 
 B
 
 
 
 
 {\displaystyle {\mathcal {B}}}
 
 satisfies the above two conditions but is not a filter base then it will form a neighborhood subbasis at 
 
 
 
 0
 
 
 {\displaystyle 0}
 
 (rather than a neighborhood basis) for a vector topology on 
 
 
 
 X
 .
 
 
 {\displaystyle X.}

In general, the set of all balanced and absorbing subsets of a vector space does not satisfy the conditions of this theorem and does not form a neighborhood basis at the origin for any vector topology.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a>

<h3>Defining topologies using strings</h3>
Let 
 
 
 
 X
 
 
 {\displaystyle X}
 
 be a vector space and let 
 
 
 
 
 U
 
 ∙
 
 
 =
 
 
 (
 
 U
 
 i
 
 
 )
 
 
 i
 =
 1
 
 
 ∞
 
 
 
 
 {\displaystyle U_{\bullet }=\left(U_{i}\right)_{i=1}^{\infty }}
 
 be a sequence of subsets of 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 
 Each set in the sequence 
 
 
 
 
 U
 
 ∙
 
 
 
 
 {\displaystyle U_{\bullet }}
 
 is called a knot of 
 
 
 
 
 U
 
 ∙
 
 
 
 
 {\displaystyle U_{\bullet }}
 
 and for every index 
 
 
 
 i
 ,
 
 
 {\displaystyle i,}
 
 
 
 
 
 
 U
 
 i
 
 
 
 
 {\displaystyle U_{i}}
 
 is called the 
 
 
 
 i
 
 
 {\displaystyle i}
 
-th knot of 
 
 
 
 
 U
 
 ∙
 
 
 .
 
 
 {\displaystyle U_{\bullet }.}
 
 The set 
 
 
 
 
 U
 
 1
 
 
 
 
 {\displaystyle U_{1}}
 
 is called the beginning of 
 
 
 
 
 U
 
 ∙
 
 
 .
 
 
 {\displaystyle U_{\bullet }.}
 
 The sequence 
 
 
 
 
 U
 
 ∙
 
 
 
 
 {\displaystyle U_{\bullet }}
 
 is/is a:<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a><a class="footnote-ref" id="fnref:12" href="#fn:12">12</a><a class="footnote-ref" id="fnref:13" href="#fn:13">13</a>

<ul><li>Summative if 
 
 
 
 
 U
 
 i
 +
 1
 
 
 +
 
 U
 
 i
 +
 1
 
 
 ⊆
 
 U
 
 i
 
 
 
 
 {\displaystyle U_{i+1}+U_{i+1}\subseteq U_{i}}
 
 for every index 
 
 
 
 i
 .
 
 
 {\displaystyle i.}
 
</li>
<li><a href="/facts/Balanced_set/Y2ar7E6g">Balanced</a> (resp. <a href="/facts/Absorbing_set/GCtjsUuX">absorbing</a>, closed,<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a> convex, open, <a href="/facts/Symmetric_set/npzCTDRx">symmetric</a>, <a href="/facts/Barrelled_space/r2H58TN3">barrelled</a>, <a href="/facts/Absolutely_convex_set/rJ5HLWkg">absolutely convex/disked</a>, etc.) if this is true of every 
 
 
 
 
 U
 
 i
 
 
 .
 
 
 {\displaystyle U_{i}.}
 
</li>
<li>String if 
 
 
 
 
 U
 
 ∙
 
 
 
 
 {\displaystyle U_{\bullet }}
 
 is summative, absorbing, and balanced.</li>
<li>Topological string or a neighborhood string in a TVS 
 
 
 
 X
 
 
 {\displaystyle X}
 
 if 
 
 
 
 
 U
 
 ∙
 
 
 
 
 {\displaystyle U_{\bullet }}
 
 is a string and each of its knots is a neighborhood of the origin in 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 
</li></ul>
If 
 
 
 
 U
 
 
 {\displaystyle U}
 
 is an <a href="/facts/Absorbing_set/GCtjsUuX">absorbing</a> <a href="/facts/Absolutely_convex_set/rJ5HLWkg">disk</a> in a vector space 
 
 
 
 X
 
 
 {\displaystyle X}
 
 then the sequence defined by 
 
 
 
 
 U
 
 i
 
 
 :=
 
 2
 
 1
 −
 i
 
 
 U
 
 
 {\displaystyle U_{i}:=2^{1-i}U}
 
 forms a string beginning with 
 
 
 
 
 U
 
 1
 
 
 =
 U
 .
 
 
 {\displaystyle U_{1}=U.}
 
 This is called the natural string of 
 
 
 
 U
 
 
 {\displaystyle U}
 
<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a> Moreover, if a vector space 
 
 
 
 X
 
 
 {\displaystyle X}
 
 has countable dimension then every string contains an <a href="/facts/Absolutely_convex_set/rJ5HLWkg">absolutely convex</a> string.
Summative sequences of sets have the particularly nice property that they define non-negative continuous real-valued <a href="/facts/Subadditive/jfRLRWfX">subadditive</a> functions. These functions can then be used to prove many of the basic properties of topological vector spaces.

Theorem (
 
 
 
 
 R
 
 
 
 {\displaystyle \mathbb {R} }
 
-valued function induced by a string)—Let 
 
 
 
 
 U
 
 ∙
 
 
 =
 
 
 (
 
 U
 
 i
 
 
 )
 
 
 i
 =
 0
 
 
 ∞
 
 
 
 
 {\displaystyle U_{\bullet }=\left(U_{i}\right)_{i=0}^{\infty }}
 
 be a collection of subsets of a vector space such that 
 
 
 
 0
 ∈
 
 U
 
 i
 
 
 
 
 {\displaystyle 0\in U_{i}}
 
 and 
 
 
 
 
 U
 
 i
 +
 1
 
 
 +
 
 U
 
 i
 +
 1
 
 
 ⊆
 
 U
 
 i
 
 
 
 
 {\displaystyle U_{i+1}+U_{i+1}\subseteq U_{i}}
 
 for all 
 
 
 
 i
 ≥
 0.
 
 
 {\displaystyle i\geq 0.}
 
 For all 
 
 
 
 u
 ∈
 
 U
 
 0
 
 
 ,
 
 
 {\displaystyle u\in U_{0},}
 
 let 
 
 
 
 
 S
 
 (
 u
 )
 :=
 
 {
 
 
 n
 
 ∙
 
 
 =
 
 (
 
 
 n
 
 1
 
 
 ,
 …
 ,
 
 n
 
 k
 
 
 
 )
 
  
 :
  
 k
 ≥
 1
 ,
 
 n
 
 i
 
 
 ≥
 0
 
  for all 
 
 i
 ,
 
  and 
 
 u
 ∈
 
 U
 
 
 n
 
 1
 
 
 
 
 +
 ⋯
 +
 
 U
 
 
 n
 
 k
 
 
 
 
 
 }
 
 .
 
 
 {\displaystyle \mathbb {S} (u):=\left\{n_{\bullet }=\left(n_{1},\ldots ,n_{k}\right)~:~k\geq 1,n_{i}\geq 0{\text{ for all }}i,{\text{ and }}u\in U_{n_{1}}+\cdots +U_{n_{k}}\right\}.}

Define 
 
 
 
 f
 :
 X
 →
 [
 0
 ,
 1
 ]
 
 
 {\displaystyle f:X\to [0,1]}
 
 by 
 
 
 
 f
 (
 x
 )
 =
 1
 
 
 {\displaystyle f(x)=1}
 
 if 
 
 
 
 x
 ∉
 
 U
 
 0
 
 
 
 
 {\displaystyle x\not \in U_{0}}
 
 and otherwise let 
 
 
 
 f
 (
 x
 )
 :=
 
 inf

{
          
            
              2
              
                −
                
                  n
                  
                    1
                  
                
              
            
            +
            ⋯
            
              2
              
                −
                
                  n
                  
                    k
                  
                
              
            
             
            :
             
            
              n
              
                ∙
              
            
            =
            
              (
              
                
                  n
                  
                    1
                  
                
                ,
                …
                ,
                
                  n
                  
                    k
                  
                
              
              )
            
            ∈
            
              S
            
            (
            x
            )
          
          }
        
        .
      
    
    {\displaystyle f(x):=\inf _{}\left\{2^{-n_{1}}+\cdots 2^{-n_{k}}~:~n_{\bullet }=\left(n_{1},\ldots ,n_{k}\right)\in \mathbb {S} (x)\right\}.}

Then 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is subadditive (meaning 
 
 
 
 f
 (
 x
 +
 y
 )
 ≤
 f
 (
 x
 )
 +
 f
 (
 y
 )
 
 
 {\displaystyle f(x+y)\leq f(x)+f(y)}
 
 for all 
 
 
 
 x
 ,
 y
 ∈
 X
 
 
 {\displaystyle x,y\in X}
 
) and 
 
 
 
 f
 =
 0
 
 
 {\displaystyle f=0}
 
 on 
 
 
 
 
 ⋂
 
 i
 ≥
 0
 
 
 
 U
 
 i
 
 
 ;
 
 
 {\textstyle \bigcap _{i\geq 0}U_{i};}
 
 so in particular, 
 
 
 
 f
 (
 0
 )
 =
 0.
 
 
 {\displaystyle f(0)=0.}
 
 If all 
 
 
 
 
 U
 
 i
 
 
 
 
 {\displaystyle U_{i}}
 
 are <a href="/facts/Symmetric_set/npzCTDRx">symmetric sets</a> then 
 
 
 
 f
 (
 −
 x
 )
 =
 f
 (
 x
 )
 
 
 {\displaystyle f(-x)=f(x)}
 
 and if all 
 
 
 
 
 U
 
 i
 
 
 
 
 {\displaystyle U_{i}}
 
 are balanced then 
 
 
 
 f
 (
 s
 x
 )
 ≤
 f
 (
 x
 )
 
 
 {\displaystyle f(sx)\leq f(x)}
 
 for all scalars 
 
 
 
 s
 
 
 {\displaystyle s}
 
 such that 
 
 
 
 
 |
 
 s
 
 |
 
 ≤
 1
 
 
 {\displaystyle |s|\leq 1}
 
 and all 
 
 
 
 x
 ∈
 X
 .
 
 
 {\displaystyle x\in X.}
 
 If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a topological vector space and if all 
 
 
 
 
 U
 
 i
 
 
 
 
 {\displaystyle U_{i}}
 
 are neighborhoods of the origin then 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is continuous, where if in addition 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is Hausdorff and 
 
 
 
 
 U
 
 ∙
 
 
 
 
 {\displaystyle U_{\bullet }}
 
 forms a basis of balanced neighborhoods of the origin in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 then 
 
 
 
 d
 (
 x
 ,
 y
 )
 :=
 f
 (
 x
 −
 y
 )
 
 
 {\displaystyle d(x,y):=f(x-y)}
 
 is a metric defining the vector topology on 
 
 
 
 X
 .
 
 
 {\displaystyle X.}

A proof of the above theorem is given in the article on <a href="/facts/Metrizable_topological_vector_space/6FUHfx6W">metrizable topological vector spaces</a>.
If 
 
 
 
 
 U
 
 ∙
 
 
 =
 
 
 (
 
 U
 
 i
 
 
 )
 
 
 i
 ∈
 
 N
 
 
 
 
 
 {\displaystyle U_{\bullet }=\left(U_{i}\right)_{i\in \mathbb {N} }}
 
 and 
 
 
 
 
 V
 
 ∙
 
 
 =
 
 
 (
 
 V
 
 i
 
 
 )
 
 
 i
 ∈
 
 N
 
 
 
 
 
 {\displaystyle V_{\bullet }=\left(V_{i}\right)_{i\in \mathbb {N} }}
 
 are two collections of subsets of a vector space 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and if 
 
 
 
 s
 
 
 {\displaystyle s}
 
 is a scalar, then by definition:<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a>

<ul><li>
 
 
 
 
 V
 
 ∙
 
 
 
 
 {\displaystyle V_{\bullet }}
 
 contains 
 
 
 
 
 U
 
 ∙
 
 
 
 
 {\displaystyle U_{\bullet }}
 
: 
 
 
 
  
 
 U
 
 ∙
 
 
 ⊆
 
 V
 
 ∙
 
 
 
 
 {\displaystyle \ U_{\bullet }\subseteq V_{\bullet }}
 
 if and only if 
 
 
 
 
 U
 
 i
 
 
 ⊆
 
 V
 
 i
 
 
 
 
 {\displaystyle U_{i}\subseteq V_{i}}
 
 for every index 
 
 
 
 i
 .
 
 
 {\displaystyle i.}
 
</li>
<li>Set of knots: 
 
 
 
  
 Knots
 ⁡
 
 U
 
 ∙
 
 
 :=
 
 {
 
 
 U
 
 i
 
 
 :
 i
 ∈
 
 N
 
 
 }
 
 .
 
 
 {\displaystyle \ \operatorname {Knots} U_{\bullet }:=\left\{U_{i}:i\in \mathbb {N} \right\}.}
 
</li>
<li>Kernel: 
 
 
 
  
 ker
 ⁡
 
 U
 
 ∙
 
 
 :=
 
 ⋂
 
 i
 ∈
 
 N
 
 
 
 
 U
 
 i
 
 
 .
 
 
 {\textstyle \ \ker U_{\bullet }:=\bigcap _{i\in \mathbb {N} }U_{i}.}
 
</li>
<li>Scalar multiple: 
 
 
 
  
 s
 
 U
 
 ∙
 
 
 :=
 
 
 (
 
 s
 
 U
 
 i
 
 
 
 )
 
 
 i
 ∈
 
 N
 
 
 
 .
 
 
 {\displaystyle \ sU_{\bullet }:=\left(sU_{i}\right)_{i\in \mathbb {N} }.}
 
</li>
<li>Sum: 
 
 
 
  
 
 U
 
 ∙
 
 
 +
 
 V
 
 ∙
 
 
 :=
 
 
 (
 
 
 U
 
 i
 
 
 +
 
 V
 
 i
 
 
 
 )
 
 
 i
 ∈
 
 N
 
 
 
 .
 
 
 {\displaystyle \ U_{\bullet }+V_{\bullet }:=\left(U_{i}+V_{i}\right)_{i\in \mathbb {N} }.}
 
</li>
<li>Intersection: 
 
 
 
  
 
 U
 
 ∙
 
 
 ∩
 
 V
 
 ∙
 
 
 :=
 
 
 (
 
 
 U
 
 i
 
 
 ∩
 
 V
 
 i
 
 
 
 )
 
 
 i
 ∈
 
 N
 
 
 
 .
 
 
 {\displaystyle \ U_{\bullet }\cap V_{\bullet }:=\left(U_{i}\cap V_{i}\right)_{i\in \mathbb {N} }.}
 
</li></ul>
If 
 
 
 
 
 S
 
 
 
 {\displaystyle \mathbb {S} }
 
 is a collection sequences of subsets of 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 then 
 
 
 
 
 S
 
 
 
 {\displaystyle \mathbb {S} }
 
 is said to be directed (downwards) under inclusion or simply directed downward if 
 
 
 
 
 S
 
 
 
 {\displaystyle \mathbb {S} }
 
 is not empty and for all 
 
 
 
 
 U
 
 ∙
 
 
 ,
 
 V
 
 ∙
 
 
 ∈
 
 S
 
 ,
 
 
 {\displaystyle U_{\bullet },V_{\bullet }\in \mathbb {S} ,}
 
 there exists some 
 
 
 
 
 W
 
 ∙
 
 
 ∈
 
 S
 
 
 
 {\displaystyle W_{\bullet }\in \mathbb {S} }
 
 such that 
 
 
 
 
 W
 
 ∙
 
 
 ⊆
 
 U
 
 ∙
 
 
 
 
 {\displaystyle W_{\bullet }\subseteq U_{\bullet }}
 
 and 
 
 
 
 
 W
 
 ∙
 
 
 ⊆
 
 V
 
 ∙
 
 
 
 
 {\displaystyle W_{\bullet }\subseteq V_{\bullet }}
 
 (said differently, if and only if 
 
 
 
 
 S
 
 
 
 {\displaystyle \mathbb {S} }
 
 is a <a href="/facts/Filter_(set_theory)/7f3zk55u">prefilter</a> with respect to the containment 
 
 
 
 
 ⊆
 
 
 
 {\displaystyle \,\subseteq \,}
 
 defined above).
Notation: Let 
 
 
 
 Knots
 ⁡
 
 S
 
 :=
 
 ⋃
 
 
 U
 
 ∙
 
 
 ∈
 
 S
 
 
 
 Knots
 ⁡
 
 U
 
 ∙
 
 
 
 
 {\textstyle \operatorname {Knots} \mathbb {S} :=\bigcup _{U_{\bullet }\in \mathbb {S} }\operatorname {Knots} U_{\bullet }}
 
 be the set of all knots of all strings in 
 
 
 
 
 S
 
 .
 
 
 {\displaystyle \mathbb {S} .}

Defining vector topologies using collections of strings is particularly useful for defining classes of TVSs that are not necessarily locally convex.

Theorem<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a> (Topology induced by strings)—If 
 
 
 
 (
 X
 ,
 τ
 )
 
 
 {\displaystyle (X,\tau )}
 
 is a topological vector space then there exists a set 
 
 
 
 
 S
 
 
 
 {\displaystyle \mathbb {S} }
 
<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a> of neighborhood strings in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 that is directed downward and such that the set of all knots of all strings in 
 
 
 
 
 S
 
 
 
 {\displaystyle \mathbb {S} }
 
 is a <a href="/facts/Neighborhood_basis/fMRM8Xht">neighborhood basis</a> at the origin for 
 
 
 
 (
 X
 ,
 τ
 )
 .
 
 
 {\displaystyle (X,\tau ).}
 
 Such a collection of strings is said to be 
 
 
 
 τ
 
 
 {\displaystyle \tau }
 
 fundamental.
Conversely, if 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a vector space and if 
 
 
 
 
 S
 
 
 
 {\displaystyle \mathbb {S} }
 
 is a collection of strings in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 that is directed downward, then the set 
 
 
 
 Knots
 ⁡
 
 S
 
 
 
 {\displaystyle \operatorname {Knots} \mathbb {S} }
 
 of all knots of all strings in 
 
 
 
 
 S
 
 
 
 {\displaystyle \mathbb {S} }
 
 forms a <a href="/facts/Neighborhood_basis/fMRM8Xht">neighborhood basis</a> at the origin for a vector topology on 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 
 In this case, this topology is denoted by 
 
 
 
 
 τ
 
 
 S
 
 
 
 
 
 {\displaystyle \tau _{\mathbb {S} }}
 
 and it is called the topology generated by 
 
 
 
 
 S
 
 .
 
 
 {\displaystyle \mathbb {S} .}

If 
 
 
 
 
 S
 
 
 
 {\displaystyle \mathbb {S} }
 
 is the set of all topological strings in a TVS 
 
 
 
 (
 X
 ,
 τ
 )
 
 
 {\displaystyle (X,\tau )}
 
 then 
 
 
 
 
 τ
 
 
 S
 
 
 
 =
 τ
 .
 
 
 {\displaystyle \tau _{\mathbb {S} }=\tau .}
 
<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a> A Hausdorff TVS is <a href="/facts/Metrizable_topological_vector_space/6FUHfx6W">metrizable</a> <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> its topology can be induced by a single topological string.<a class="footnote-ref" id="fnref:20" href="#fn:20">20</a>

<h2 id="topological-structure">Topological structure</h2>
A vector space is an <a href="/facts/Abelian_group/FfWwmx4R">abelian group</a> with respect to the operation of addition, and in a topological vector space the inverse operation is always continuous (since it is the same as multiplication by 
 
 
 
 −
 1
 
 
 {\displaystyle -1}
 
). Hence, every topological vector space is an abelian <a href="/facts/Topological_group/cAgNMMRU">topological group</a>. Every TVS is <a href="/facts/Completely_regular/jyCgpKB5">completely regular</a> but a TVS need not be <a href="/facts/Normal_space/t4DLqLCd">normal</a>.<a class="footnote-ref" id="fnref:21" href="#fn:21">21</a>
Let 
 
 
 
 X
 
 
 {\displaystyle X}
 
 be a topological vector space. Given a <a href="/facts/Subspace_topology/NjU6z4Ca">subspace</a> 
 
 
 
 M
 ⊆
 X
 ,
 
 
 {\displaystyle M\subseteq X,}
 
 the quotient space 
 
 
 
 X
 
 /
 
 M
 
 
 {\displaystyle X/M}
 
 with the usual <a href="/facts/Quotient_space_(topology)/muE0qQVC">quotient topology</a> is a Hausdorff topological vector space if and only if 
 
 
 
 M
 
 
 {\displaystyle M}
 
 is closed.<a class="footnote-ref" id="fnref:22" href="#fn:22">22</a> This permits the following construction: given a topological vector space 
 
 
 
 X
 
 
 {\displaystyle X}
 
 (that is probably not Hausdorff), form the quotient space 
 
 
 
 X
 
 /
 
 M
 
 
 {\displaystyle X/M}
 
 where 
 
 
 
 M
 
 
 {\displaystyle M}
 
 is the closure of 
 
 
 
 {
 0
 }
 .
 
 
 {\displaystyle \{0\}.}
 
 
 
 
 
 X
 
 /
 
 M
 
 
 {\displaystyle X/M}
 
 is then a Hausdorff topological vector space that can be studied instead of 
 
 
 
 X
 .
 
 
 {\displaystyle X.}

<h3>Invariance of vector topologies</h3>
One of the most used properties of vector topologies is that every vector topology is translation invariant:

for all 
 
 
 
 
 x
 
 0
 
 
 ∈
 X
 ,
 
 
 {\displaystyle x_{0}\in X,}
 
 the map 
 
 
 
 X
 →
 X
 
 
 {\displaystyle X\to X}
 
 defined by 
 
 
 
 x
 ↦
 
 x
 
 0
 
 
 +
 x
 
 
 {\displaystyle x\mapsto x_{0}+x}
 
 is a <a href="/facts/Homeomorphism/jMfWg3Mn">homeomorphism</a>, but if 
 
 
 
 
 x
 
 0
 
 
 ≠
 0
 
 
 {\displaystyle x_{0}\neq 0}
 
 then it is not linear and so not a TVS-isomorphism.
Scalar multiplication by a non-zero scalar is a TVS-isomorphism. This means that if 
 
 
 
 s
 ≠
 0
 
 
 {\displaystyle s\neq 0}
 
 then the linear map 
 
 
 
 X
 →
 X
 
 
 {\displaystyle X\to X}
 
 defined by 
 
 
 
 x
 ↦
 s
 x
 
 
 {\displaystyle x\mapsto sx}
 
 is a homeomorphism. Using 
 
 
 
 s
 =
 −
 1
 
 
 {\displaystyle s=-1}
 
 produces the negation map 
 
 
 
 X
 →
 X
 
 
 {\displaystyle X\to X}
 
 defined by 
 
 
 
 x
 ↦
 −
 x
 ,
 
 
 {\displaystyle x\mapsto -x,}
 
 which is consequently a linear homeomorphism and thus a TVS-isomorphism.
If 
 
 
 
 x
 ∈
 X
 
 
 {\displaystyle x\in X}
 
 and any subset 
 
 
 
 S
 ⊆
 X
 ,
 
 
 {\displaystyle S\subseteq X,}
 
 then 
 
 
 
 
 cl
 
 X
 
 
 ⁡
 (
 x
 +
 S
 )
 =
 x
 +
 
 cl
 
 X
 
 
 ⁡
 S
 
 
 {\displaystyle \operatorname {cl} _{X}(x+S)=x+\operatorname {cl} _{X}S}
 
<a class="footnote-ref" id="fnref:23" href="#fn:23">23</a> and moreover, if 
 
 
 
 0
 ∈
 S
 
 
 {\displaystyle 0\in S}
 
 then 
 
 
 
 x
 +
 S
 
 
 {\displaystyle x+S}
 
 is a <a href="/facts/Neighborhood_(topology)/XHKgrpkp">neighborhood</a> (resp. open neighborhood, closed neighborhood) of 
 
 
 
 x
 
 
 {\displaystyle x}
 
 in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 if and only if the same is true of 
 
 
 
 S
 
 
 {\displaystyle S}
 
 at the origin.

<h3>Local notions</h3>
A subset 
 
 
 
 E
 
 
 {\displaystyle E}
 
 of a vector space 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is said to be

<ul><li><a href="/facts/Absorbing_set/GCtjsUuX">absorbing</a> (in 
 
 
 
 X
 
 
 {\displaystyle X}
 
): if for every 
 
 
 
 x
 ∈
 X
 ,
 
 
 {\displaystyle x\in X,}
 
 there exists a real 
 
 
 
 r
 >
 0
 
 
 {\displaystyle r>0}
 
 such that 
 
 
 
 c
 x
 ∈
 E
 
 
 {\displaystyle cx\in E}
 
 for any scalar 
 
 
 
 c
 
 
 {\displaystyle c}
 
 satisfying 
 
 
 
 
 |
 
 c
 
 |
 
 ≤
 r
 .
 
 
 {\displaystyle |c|\leq r.}
 
<a class="footnote-ref" id="fnref:24" href="#fn:24">24</a></li>
<li><a href="/facts/Balanced_set/Y2ar7E6g">balanced</a> or circled: if 
 
 
 
 t
 E
 ⊆
 E
 
 
 {\displaystyle tE\subseteq E}
 
 for every scalar 
 
 
 
 
 |
 
 t
 
 |
 
 ≤
 1.
 
 
 {\displaystyle |t|\leq 1.}
 
<a class="footnote-ref" id="fnref:25" href="#fn:25">25</a></li>
<li><a href="/facts/Convex_set/vdAuJRJl">convex</a>: if 
 
 
 
 t
 E
 +
 (
 1
 −
 t
 )
 E
 ⊆
 E
 
 
 {\displaystyle tE+(1-t)E\subseteq E}
 
 for every real 
 
 
 
 0
 ≤
 t
 ≤
 1.
 
 
 {\displaystyle 0\leq t\leq 1.}
 
<a class="footnote-ref" id="fnref:26" href="#fn:26">26</a></li>
<li>a <a href="/facts/Absolutely_convex_set/rJ5HLWkg">disk</a> or <a href="/facts/Absolutely_convex_set/rJ5HLWkg">absolutely convex</a>: if 
 
 
 
 E
 
 
 {\displaystyle E}
 
 is convex and balanced.</li>
<li><a href="/facts/Symmetric_set/npzCTDRx">symmetric</a>: if 
 
 
 
 −
 E
 ⊆
 E
 ,
 
 
 {\displaystyle -E\subseteq E,}
 
 or equivalently, if 
 
 
 
 −
 E
 =
 E
 .
 
 
 {\displaystyle -E=E.}
 
</li></ul>
Every neighborhood of the origin is an <a href="/facts/Absorbing_set/GCtjsUuX">absorbing set</a> and contains an open <a href="/facts/Balanced_set/Y2ar7E6g">balanced</a> neighborhood of 
 
 
 
 0
 
 
 {\displaystyle 0}
 
<a class="footnote-ref" id="fnref:27" href="#fn:27">27</a> so every topological vector space has a local base of absorbing and <a href="/facts/Balanced_set/Y2ar7E6g">balanced sets</a>. The origin even has a neighborhood basis consisting of closed balanced neighborhoods of 
 
 
 
 0
 ;
 
 
 {\displaystyle 0;}
 
 if the space is <a href="/facts/Locally_convex/AGrPrayC">locally convex</a> then it also has a neighborhood basis consisting of closed convex balanced neighborhoods of the origin.
Bounded subsets
A subset 
 
 
 
 E
 
 
 {\displaystyle E}
 
 of a topological vector space 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is <a href="/facts/Bounded_set_(topological_vector_space)/bTsWeU9g">bounded</a><a class="footnote-ref" id="fnref:28" href="#fn:28">28</a> if for every neighborhood 
 
 
 
 V
 
 
 {\displaystyle V}
 
 of the origin there exists 
 
 
 
 t
 
 
 {\displaystyle t}
 
 such that 
 
 
 
 E
 ⊆
 t
 V
 
 
 {\displaystyle E\subseteq tV}
 
.
The definition of boundedness can be weakened a bit; 
 
 
 
 E
 
 
 {\displaystyle E}
 
 is bounded if and only if every countable subset of it is bounded. A set is bounded if and only if each of its subsequences is a bounded set.<a class="footnote-ref" id="fnref:29" href="#fn:29">29</a> Also, 
 
 
 
 E
 
 
 {\displaystyle E}
 
 is bounded if and only if for every balanced neighborhood 
 
 
 
 V
 
 
 {\displaystyle V}
 
 of the origin, there exists 
 
 
 
 t
 
 
 {\displaystyle t}
 
 such that 
 
 
 
 E
 ⊆
 t
 V
 .
 
 
 {\displaystyle E\subseteq tV.}
 
 Moreover, when 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is locally convex, the boundedness can be characterized by <a href="/facts/Seminorm/vMuescKp">seminorms</a>: the subset 
 
 
 
 E
 
 
 {\displaystyle E}
 
 is bounded if and only if every continuous seminorm 
 
 
 
 p
 
 
 {\displaystyle p}
 
 is bounded on 
 
 
 
 E
 .
 
 
 {\displaystyle E.}
 
<a class="footnote-ref" id="fnref:30" href="#fn:30">30</a>
Every <a href="/facts/Totally_bounded/H8GSU1TT">totally bounded</a> set is bounded.<a class="footnote-ref" id="fnref:31" href="#fn:31">31</a> If 
 
 
 
 M
 
 
 {\displaystyle M}
 
 is a vector subspace of a TVS 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 then a subset of 
 
 
 
 M
 
 
 {\displaystyle M}
 
 is bounded in 
 
 
 
 M
 
 
 {\displaystyle M}
 
 if and only if it is bounded in 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 
<a class="footnote-ref" id="fnref:32" href="#fn:32">32</a>

<h3>Metrizability</h3>

<a href="/facts/Birkhoff%25E2%2580%2593Kakutani_theorem/cAgNMMRU">Birkhoff–Kakutani theorem</a>—If 
 
 
 
 (
 X
 ,
 τ
 )
 
 
 {\displaystyle (X,\tau )}
 
 is a topological vector space then the following four conditions are equivalent:<a class="footnote-ref" id="fnref:33" href="#fn:33">33</a><a class="footnote-ref" id="fnref:34" href="#fn:34">34</a>

<ol><li>The origin 
 
 
 
 {
 0
 }
 
 
 {\displaystyle \{0\}}
 
 is closed in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and there is a <a href="/facts/Countable/Wj4DmWPv">countable</a> <a href="/facts/Neighborhood_basis/fMRM8Xht">basis of neighborhoods</a> at the origin in 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 
</li>
<li>
 
 
 
 (
 X
 ,
 τ
 )
 
 
 {\displaystyle (X,\tau )}
 
 is <a href="/facts/Metrizable_space/BCLdo5Jf">metrizable</a> (as a topological space).</li>
<li>There is a <a href="/facts/Translation-invariant_metric/RkUptSo8">translation-invariant metric</a> on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 that induces on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 the topology 
 
 
 
 τ
 ,
 
 
 {\displaystyle \tau ,}
 
 which is the given topology on 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 
</li>
<li>
 
 
 
 (
 X
 ,
 τ
 )
 
 
 {\displaystyle (X,\tau )}
 
 is a <a href="/facts/Metrizable_topological_vector_space/6FUHfx6W">metrizable topological vector space</a>.<a class="footnote-ref" id="fnref:35" href="#fn:35">35</a></li></ol>
By the Birkhoff–Kakutani theorem, it follows that there is an <a href="/facts/Equivalence_of_metrics/kezfD0T4">equivalent metric</a> that is translation-invariant.

A TVS is <a href="/facts/Metrizable_TVS/6FUHfx6W">pseudometrizable</a> if and only if it has a countable neighborhood basis at the origin, or equivalent, if and only if its topology is generated by an <a href="/facts/Metrizable_TVS/6FUHfx6W">F-seminorm</a>. A TVS is metrizable if and only if it is Hausdorff and pseudometrizable.
More strongly: a topological vector space is said to be <a href="/facts/Normable/xIbR4uE1">normable</a> if its topology can be induced by a norm. A topological vector space is normable if and only if it is Hausdorff and has a convex bounded neighborhood of the origin.<a class="footnote-ref" id="fnref:36" href="#fn:36">36</a>
Let 
 
 
 
 
 K
 
 
 
 {\displaystyle \mathbb {K} }
 
 be a non-<a href="/facts/Discrete_space/hpvXA23u">discrete</a> <a href="/facts/Locally_compact/hesalT7q">locally compact</a> topological field, for example the real or complex numbers. A <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff</a> topological vector space over 
 
 
 
 
 K
 
 
 
 {\displaystyle \mathbb {K} }
 
 is locally compact if and only if it is <a href="/facts/Finite-dimensional/z8m02A3W">finite-dimensional</a>, that is, isomorphic to 
 
 
 
 
 
 K
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {K} ^{n}}
 
 for some natural number 
 
 
 
 n
 .
 
 
 {\displaystyle n.}
 
<a class="footnote-ref" id="fnref:37" href="#fn:37">37</a>

<h3>Completeness and uniform structure</h3>
Main article: <a href="/facts/Complete_topological_vector_space/e6qtS76j">Complete topological vector space</a>
The <a href="/facts/Complete_topological_vector_space/e6qtS76j">canonical uniformity</a><a class="footnote-ref" id="fnref:38" href="#fn:38">38</a> on a TVS 
 
 
 
 (
 X
 ,
 τ
 )
 
 
 {\displaystyle (X,\tau )}
 
 is the unique translation-invariant <a href="/facts/Uniform_space/GKumHlHo">uniformity</a> that induces the topology 
 
 
 
 τ
 
 
 {\displaystyle \tau }
 
 on 
 
 
 
 X
 .
 
 
 {\displaystyle X.}

Every TVS is assumed to be endowed with this canonical uniformity, which makes all TVSs into <a href="/facts/Uniform_space/GKumHlHo">uniform spaces</a>. This allows one to talk about related notions such as <a href="/facts/Complete_topological_vector_space/e6qtS76j">completeness</a>, <a href="/facts/Uniform_convergence/ecz9eNWn">uniform convergence</a>, Cauchy nets, and <a href="/facts/Uniform_continuity/hjvw7tII">uniform continuity</a>, etc., which are always assumed to be with respect to this uniformity (unless indicated other). This implies that every Hausdorff topological vector space is <a href="/facts/Tychonoff_space/jyCgpKB5">Tychonoff</a>.<a class="footnote-ref" id="fnref:39" href="#fn:39">39</a> A subset of a TVS is <a href="/facts/Compact_space/d0cgXJH7">compact</a> if and only if it is complete and <a href="/facts/Totally_bounded/H8GSU1TT">totally bounded</a> (for Hausdorff TVSs, a set being totally bounded is equivalent to it being <a href="/facts/Totally_bounded_space/H8GSU1TT">precompact</a>). But if the TVS is not Hausdorff then there exist compact subsets that are not closed. However, the closure of a compact subset of a non-Hausdorff TVS is again compact (so compact subsets are <a href="/facts/Relatively_compact/z5uIqMDp">relatively compact</a>).
With respect to this uniformity, a <a href="/facts/Net_(mathematics)/msmsvMTT">net</a> (or <a href="/facts/Sequence_(mathematics)/gV2S4uqp">sequence</a>) 
 
 
 
 
 x
 
 ∙
 
 
 =
 
 
 (
 
 x
 
 i
 
 
 )
 
 
 i
 ∈
 I
 
 
 
 
 {\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}
 
 is Cauchy if and only if for every neighborhood 
 
 
 
 V
 
 
 {\displaystyle V}
 
 of 
 
 
 
 0
 ,
 
 
 {\displaystyle 0,}
 
 there exists some index 
 
 
 
 n
 
 
 {\displaystyle n}
 
 such that 
 
 
 
 
 x
 
 i
 
 
 −
 
 x
 
 j
 
 
 ∈
 V
 
 
 {\displaystyle x_{i}-x_{j}\in V}
 
 whenever 
 
 
 
 i
 ≥
 n
 
 
 {\displaystyle i\geq n}
 
 and 
 
 
 
 j
 ≥
 n
 .
 
 
 {\displaystyle j\geq n.}

Every <a href="/facts/Cauchy_sequence/7Q5c8TuF">Cauchy sequence</a> is bounded, although Cauchy nets and Cauchy filters may not be bounded. A topological vector space where every Cauchy sequence converges is called <a href="/facts/Sequentially_complete/8avaV5Uu">sequentially complete</a>; in general, it may not be complete (in the sense that all Cauchy filters converge).
The vector space operation of addition is uniformly continuous and an open map. Scalar multiplication is <a href="/facts/Cauchy_continuous/PdajK5yx">Cauchy continuous</a> but in general, it is almost never uniformly continuous. Because of this, every topological vector space can be completed and is thus a <a href="/facts/Dense_set/nPT9Yxao">dense</a> <a href="/facts/Linear_subspace/2wtKTgHL">linear subspace</a> of a <a href="/facts/Complete_topological_vector_space/e6qtS76j">complete topological vector space</a>.

<ul><li>Every TVS has a <a href="/facts/Complete_topological_vector_space/e6qtS76j">completion</a> and every Hausdorff TVS has a Hausdorff completion.<a class="footnote-ref" id="fnref:40" href="#fn:40">40</a> Every TVS (even those that are Hausdorff and/or complete) has infinitely many non-isomorphic non-Hausdorff completions.</li>
<li>A compact subset of a TVS (not necessarily Hausdorff) is complete.<a class="footnote-ref" id="fnref:41" href="#fn:41">41</a> A complete subset of a Hausdorff TVS is closed.<a class="footnote-ref" id="fnref:42" href="#fn:42">42</a></li>
<li>If 
 
 
 
 C
 
 
 {\displaystyle C}
 
 is a complete subset of a TVS then any subset of 
 
 
 
 C
 
 
 {\displaystyle C}
 
 that is closed in 
 
 
 
 C
 
 
 {\displaystyle C}
 
 is complete.<a class="footnote-ref" id="fnref:43" href="#fn:43">43</a></li>
<li>A Cauchy sequence in a Hausdorff TVS 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is not necessarily <a href="/facts/Relatively_compact/z5uIqMDp">relatively compact</a> (that is, its closure in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is not necessarily compact).</li>
<li>If a Cauchy filter in a TVS has an <a href="/facts/Filters_in_topology/5vQbHcpE">accumulation point</a> 
 
 
 
 x
 
 
 {\displaystyle x}
 
 then it converges to 
 
 
 
 x
 .
 
 
 {\displaystyle x.}
 
</li>
<li>If a series 
 
 
 
 
 ∑
 
 i
 =
 1
 
 
 ∞
 
 
 
 x
 
 i
 
 
 
 
 {\textstyle \sum _{i=1}^{\infty }x_{i}}
 
 converges<a class="footnote-ref" id="fnref:44" href="#fn:44">44</a> in a TVS 
 
 
 
 X
 
 
 {\displaystyle X}
 
 then 
 
 
 
 
 x
 
 ∙
 
 
 →
 0
 
 
 {\displaystyle x_{\bullet }\to 0}
 
 in 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 
<a class="footnote-ref" id="fnref:45" href="#fn:45">45</a></li></ul>
<h2 id="examples">Examples</h2>
<h3>Finest and coarsest vector topology</h3>
Let 
 
 
 
 X
 
 
 {\displaystyle X}
 
 be a real or complex vector space.
Trivial topology
The <a href="/facts/Trivial_topology/uA7RlGth">trivial topology</a> or indiscrete topology 
 
 
 
 {
 X
 ,
 ∅
 }
 
 
 {\displaystyle \{X,\varnothing \}}
 
 is always a TVS topology on any vector space 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and it is the coarsest TVS topology possible. An important consequence of this is that the intersection of any collection of TVS topologies on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 always contains a TVS topology. Any vector space (including those that are infinite dimensional) endowed with the trivial topology is a compact (and thus <a href="/facts/Locally_compact/hesalT7q">locally compact</a>) <a href="/facts/Complete_topological_vector_space/e6qtS76j">complete</a> <a href="/facts/Metrizable_topological_vector_space/6FUHfx6W">pseudometrizable</a> <a href="/facts/Seminormed_space/vMuescKp">seminormable</a> <a href="/facts/Locally_convex_topological_vector_space/AGrPrayC">locally convex</a> topological vector space. It is <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff</a> if and only if 
 
 
 
 dim
 ⁡
 X
 =
 0.
 
 
 {\displaystyle \dim X=0.}

Finest vector topology
There exists a TVS topology 
 
 
 
 
 τ
 
 f
 
 
 
 
 {\displaystyle \tau _{f}}
 
 on 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 called the finest vector topology on 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 that is finer than every other TVS-topology on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 (that is, any TVS-topology on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is necessarily a subset of 
 
 
 
 
 τ
 
 f
 
 
 
 
 {\displaystyle \tau _{f}}
 
).<a class="footnote-ref" id="fnref:46" href="#fn:46">46</a><a class="footnote-ref" id="fnref:47" href="#fn:47">47</a> Every linear map from 
 
 
 
 
 (
 
 X
 ,
 
 τ
 
 f
 
 
 
 )
 
 
 
 {\displaystyle \left(X,\tau _{f}\right)}
 
 into another TVS is necessarily continuous. If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 has an uncountable <a href="/facts/Hamel_basis/89IPoN6c">Hamel basis</a> then 
 
 
 
 
 τ
 
 f
 
 
 
 
 {\displaystyle \tau _{f}}
 
 is not <a href="/facts/Locally_convex_topological_vector_space/AGrPrayC">locally convex</a> and not <a href="/facts/Metrizable_topological_vector_space/6FUHfx6W">metrizable</a>.<a class="footnote-ref" id="fnref:48" href="#fn:48">48</a>

<h3>Cartesian products</h3>
A <a href="/facts/Cartesian_product/BfqBWzdL">Cartesian product</a> of a family of topological vector spaces, when endowed with the <a href="/facts/Product_topology/0hg02IaX">product topology</a>, is a topological vector space. Consider for instance the set 
 
 
 
 X
 
 
 {\displaystyle X}
 
 of all functions 
 
 
 
 f
 :
 
 R
 
 →
 
 R
 
 
 
 {\displaystyle f:\mathbb {R} \to \mathbb {R} }
 
 where 
 
 
 
 
 R
 
 
 
 {\displaystyle \mathbb {R} }
 
 carries its usual <a href="/facts/Euclidean_topology/gCcjWJvA">Euclidean topology</a>. This set 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a real vector space (where addition and scalar multiplication are defined pointwise, as usual) that can be identified with (and indeed, is often defined to be) the <a href="/facts/Cartesian_product/BfqBWzdL">Cartesian product</a> 
 
 
 
 
 
 R
 
 
 
 R
 
 
 
 ,
 ,
 
 
 {\displaystyle \mathbb {R} ^{\mathbb {R} },,}
 
 which carries the natural <a href="/facts/Product_topology/0hg02IaX">product topology</a>. With this product topology, 
 
 
 
 X
 :=
 
 
 R
 
 
 
 R
 
 
 
 
 
 {\displaystyle X:=\mathbb {R} ^{\mathbb {R} }}
 
 becomes a topological vector space whose topology is called the topology of <a href="/facts/Pointwise_convergence/TXGpeyIK">pointwise convergence</a> on 
 
 
 
 
 R
 
 .
 
 
 {\displaystyle \mathbb {R} .}
 
 The reason for this name is the following: if 
 
 
 
 
 
 (
 
 f
 
 n
 
 
 )
 
 
 n
 =
 1
 
 
 ∞
 
 
 
 
 {\displaystyle \left(f_{n}\right)_{n=1}^{\infty }}
 
 is a <a href="/facts/Sequence/gV2S4uqp">sequence</a> (or more generally, a <a href="/facts/Net_(mathematics)/msmsvMTT">net</a>) of elements in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and if 
 
 
 
 f
 ∈
 X
 
 
 {\displaystyle f\in X}
 
 then 
 
 
 
 
 f
 
 n
 
 
 
 
 {\displaystyle f_{n}}
 
 <a href="/facts/Limit_of_a_sequence/Ktge2RwL">converges</a> to 
 
 
 
 f
 
 
 {\displaystyle f}
 
 in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 if and only if for every real number 
 
 
 
 x
 ,
 
 
 {\displaystyle x,}
 
 
 
 
 
 
 f
 
 n
 
 
 (
 x
 )
 
 
 {\displaystyle f_{n}(x)}
 
 converges to 
 
 
 
 f
 (
 x
 )
 
 
 {\displaystyle f(x)}
 
 in 
 
 
 
 
 R
 
 .
 
 
 {\displaystyle \mathbb {R} .}
 
 This TVS is <a href="/facts/Complete_topological_vector_space/e6qtS76j">complete</a>, <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff</a>, and <a href="/facts/Locally_convex/AGrPrayC">locally convex</a> but not <a href="/facts/Metrizable_topological_vector_space/6FUHfx6W">metrizable</a> and consequently not <a href="/facts/Normable/xIbR4uE1">normable</a>; indeed, every neighborhood of the origin in the product topology contains lines (that is, 1-dimensional vector subspaces, which are subsets of the form 
 
 
 
 
 R
 
 f
 :=
 {
 r
 f
 :
 r
 ∈
 
 R
 
 }
 
 
 {\displaystyle \mathbb {R} f:=\{rf:r\in \mathbb {R} \}}
 
 with 
 
 
 
 f
 ≠
 0
 
 
 {\displaystyle f\neq 0}
 
).

<h3>Finite-dimensional spaces</h3>
By <a href="/facts/F._Riesz%2527s_theorem/esv1y0Xj">F. Riesz's theorem</a>, a Hausdorff topological vector space is finite-dimensional if and only if it is <a href="/facts/Locally_compact/hesalT7q">locally compact</a>, which happens if and only if it has a compact <a href="/facts/Neighborhood_(topology)/XHKgrpkp">neighborhood</a> of the origin.
Let 
 
 
 
 
 K
 
 
 
 {\displaystyle \mathbb {K} }
 
 denote 
 
 
 
 
 R
 
 
 
 {\displaystyle \mathbb {R} }
 
 or 
 
 
 
 
 C
 
 
 
 {\displaystyle \mathbb {C} }
 
 and endow 
 
 
 
 
 K
 
 
 
 {\displaystyle \mathbb {K} }
 
 with its usual Hausdorff normed <a href="/facts/Euclidean_topology/gCcjWJvA">Euclidean topology</a>. Let 
 
 
 
 X
 
 
 {\displaystyle X}
 
 be a vector space over 
 
 
 
 
 K
 
 
 
 {\displaystyle \mathbb {K} }
 
 of finite dimension 
 
 
 
 n
 :=
 dim
 ⁡
 X
 
 
 {\displaystyle n:=\dim X}
 
 and so that 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is vector space isomorphic to 
 
 
 
 
 
 K
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {K} ^{n}}
 
 (explicitly, this means that there exists a <a href="/facts/Linear_isomorphism/Q1PBKNVV">linear isomorphism</a> between the vector spaces 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 
 
 K
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {K} ^{n}}
 
). This finite-dimensional vector space 
 
 
 
 X
 
 
 {\displaystyle X}
 
 always has a unique <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff</a> vector topology, which makes it TVS-isomorphic to 
 
 
 
 
 
 K
 
 
 n
 
 
 ,
 
 
 {\displaystyle \mathbb {K} ^{n},}
 
 where 
 
 
 
 
 
 K
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {K} ^{n}}
 
 is endowed with the usual Euclidean topology (which is the same as the <a href="/facts/Product_topology/0hg02IaX">product topology</a>). This Hausdorff vector topology is also the (unique) <a href="/facts/Comparison_of_topologies/5Q9F2fjf">finest</a> vector topology on 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 
 
 
 
 
 X
 
 
 {\displaystyle X}
 
 has a unique vector topology if and only if 
 
 
 
 dim
 ⁡
 X
 =
 0.
 
 
 {\displaystyle \dim X=0.}
 
 If 
 
 
 
 dim
 ⁡
 X
 ≠
 0
 
 
 {\displaystyle \dim X\neq 0}
 
 then although 
 
 
 
 X
 
 
 {\displaystyle X}
 
 does not have a unique vector topology, it does have a unique Hausdorff vector topology.

<ul><li>If 
 
 
 
 dim
 ⁡
 X
 =
 0
 
 
 {\displaystyle \dim X=0}
 
 then 
 
 
 
 X
 =
 {
 0
 }
 
 
 {\displaystyle X=\{0\}}
 
 has exactly one vector topology: the <a href="/facts/Trivial_topology/uA7RlGth">trivial topology</a>, which in this case (and only in this case) is Hausdorff. The trivial topology on a vector space is Hausdorff if and only if the vector space has dimension 
 
 
 
 0.
 
 
 {\displaystyle 0.}
 
</li>
<li>If 
 
 
 
 dim
 ⁡
 X
 =
 1
 
 
 {\displaystyle \dim X=1}
 
 then 
 
 
 
 X
 
 
 {\displaystyle X}
 
 has two vector topologies: the usual <a href="/facts/Euclidean_topology/gCcjWJvA">Euclidean topology</a> and the (non-Hausdorff) trivial topology.
<ul><li>Since the field 
 
 
 
 
 K
 
 
 
 {\displaystyle \mathbb {K} }
 
 is itself a 
 
 
 
 1
 
 
 {\displaystyle 1}
 
-dimensional topological vector space over 
 
 
 
 
 K
 
 
 
 {\displaystyle \mathbb {K} }
 
 and since it plays an important role in the definition of topological vector spaces, this dichotomy plays an important role in the definition of an <a href="/facts/Absorbing_set/GCtjsUuX">absorbing set</a> and has consequences that reverberate throughout <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a>.</li></ul></li></ul>
Proof outline
The proof of this dichotomy (i.e. that a vector topology is either trivial or isomorphic to 
 
 
 
 
 K
 
 
 
 {\displaystyle \mathbb {K} }
 
) is straightforward so only an outline with the important observations is given. As usual, 
 
 
 
 
 K
 
 
 
 {\displaystyle \mathbb {K} }
 
 is assumed have the (normed) Euclidean topology. Let 
 
 
 
 
 B
 
 r
 
 
 :=
 {
 a
 ∈
 
 K
 
 :
 
 |
 
 a
 
 |
 
 <
 r
 }
 
 
 {\displaystyle B_{r}:=\{a\in \mathbb {K} :|a|<r\}}
 
 for all 
 
 
 
 r
 >
 0.
 
 
 {\displaystyle r>0.}
 
 Let 
 
 
 
 X
 
 
 {\displaystyle X}
 
 be a 
 
 
 
 1
 
 
 {\displaystyle 1}
 
-dimensional vector space over 
 
 
 
 
 K
 
 .
 
 
 {\displaystyle \mathbb {K} .}
 
 If 
 
 
 
 S
 ⊆
 X
 
 
 {\displaystyle S\subseteq X}
 
 and 
 
 
 
 B
 ⊆
 
 K
 
 
 
 {\displaystyle B\subseteq \mathbb {K} }
 
 is a ball centered at 
 
 
 
 0
 
 
 {\displaystyle 0}
 
 then 
 
 
 
 B
 ⋅
 S
 =
 X
 
 
 {\displaystyle B\cdot S=X}
 
 whenever 
 
 
 
 S
 
 
 {\displaystyle S}
 
 contains an "unbounded sequence", by which it is meant a sequence of the form 
 
 
 
 
 
 (
 
 
 a
 
 i
 
 
 x
 
 )
 
 
 i
 =
 1
 
 
 ∞
 
 
 
 
 {\displaystyle \left(a_{i}x\right)_{i=1}^{\infty }}
 
 where 
 
 
 
 0
 ≠
 x
 ∈
 X
 
 
 {\displaystyle 0\neq x\in X}
 
 and 
 
 
 
 
 
 (
 
 a
 
 i
 
 
 )
 
 
 i
 =
 1
 
 
 ∞
 
 
 ⊆
 
 K
 
 
 
 {\displaystyle \left(a_{i}\right)_{i=1}^{\infty }\subseteq \mathbb {K} }
 
 is unbounded in normed space 
 
 
 
 
 K
 
 
 
 {\displaystyle \mathbb {K} }
 
 (in the usual sense). Any vector topology on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 will be translation invariant and invariant under non-zero scalar multiplication, and for every 
 
 
 
 0
 ≠
 x
 ∈
 X
 ,
 
 
 {\displaystyle 0\neq x\in X,}
 
 the map 
 
 
 
 
 M
 
 x
 
 
 :
 
 K
 
 →
 X
 
 
 {\displaystyle M_{x}:\mathbb {K} \to X}
 
 given by 
 
 
 
 
 M
 
 x
 
 
 (
 a
 )
 :=
 a
 x
 
 
 {\displaystyle M_{x}(a):=ax}
 
 is a continuous linear bijection. Because 
 
 
 
 X
 =
 
 K
 
 x
 
 
 {\displaystyle X=\mathbb {K} x}
 
 for any such 
 
 
 
 x
 ,
 
 
 {\displaystyle x,}
 
 every subset of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 can be written as 
 
 
 
 F
 x
 =
 
 M
 
 x
 
 
 (
 F
 )
 
 
 {\displaystyle Fx=M_{x}(F)}
 
 for some unique subset 
 
 
 
 F
 ⊆
 
 K
 
 .
 
 
 {\displaystyle F\subseteq \mathbb {K} .}
 
 And if this vector topology on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 has a neighborhood 
 
 
 
 W
 
 
 {\displaystyle W}
 
 of the origin that is not equal to all of 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 then the continuity of scalar multiplication 
 
 
 
 
 K
 
 ×
 X
 →
 X
 
 
 {\displaystyle \mathbb {K} \times X\to X}
 
 at the origin guarantees the existence of an open ball 
 
 
 
 
 B
 
 r
 
 
 ⊆
 
 K
 
 
 
 {\displaystyle B_{r}\subseteq \mathbb {K} }
 
 centered at 
 
 
 
 0
 
 
 {\displaystyle 0}
 
 and an open neighborhood 
 
 
 
 S
 
 
 {\displaystyle S}
 
 of the origin in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 such that 
 
 
 
 
 B
 
 r
 
 
 ⋅
 S
 ⊆
 W
 ≠
 X
 ,
 
 
 {\displaystyle B_{r}\cdot S\subseteq W\neq X,}
 
 which implies that 
 
 
 
 S
 
 
 {\displaystyle S}
 
 does not contain any "unbounded sequence". This implies that for every 
 
 
 
 0
 ≠
 x
 ∈
 X
 ,
 
 
 {\displaystyle 0\neq x\in X,}
 
 there exists some positive integer 
 
 
 
 n
 
 
 {\displaystyle n}
 
 such that 
 
 
 
 S
 ⊆
 
 B
 
 n
 
 
 x
 .
 
 
 {\displaystyle S\subseteq B_{n}x.}
 
 From this, it can be deduced that if 
 
 
 
 X
 
 
 {\displaystyle X}
 
 does not carry the trivial topology and if 
 
 
 
 0
 ≠
 x
 ∈
 X
 ,
 
 
 {\displaystyle 0\neq x\in X,}
 
 then for any ball 
 
 
 
 B
 ⊆
 
 K
 
 
 
 {\displaystyle B\subseteq \mathbb {K} }
 
 center at 0 in 
 
 
 
 
 K
 
 ,
 
 
 {\displaystyle \mathbb {K} ,}
 
 
 
 
 
 
 M
 
 x
 
 
 (
 B
 )
 =
 B
 x
 
 
 {\displaystyle M_{x}(B)=Bx}
 
 contains an open neighborhood of the origin in 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 which then proves that 
 
 
 
 
 M
 
 x
 
 
 
 
 {\displaystyle M_{x}}
 
 is a linear <a href="/facts/Homeomorphism/jMfWg3Mn">homeomorphism</a>. <a href="/facts/Q.E.D./B8I8jPwT">Q.E.D.</a> 
 
 
 
 ◼
 
 
 {\displaystyle \blacksquare }

<ul><li>If 
 
 
 
 dim
 ⁡
 X
 =
 n
 ≥
 2
 
 
 {\displaystyle \dim X=n\geq 2}
 
 then 
 
 
 
 X
 
 
 {\displaystyle X}
 
 has infinitely many distinct vector topologies:
<ul><li>Some of these topologies are now described: Every linear functional 
 
 
 
 f
 
 
 {\displaystyle f}
 
 on 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 which is vector space isomorphic to 
 
 
 
 
 
 K
 
 
 n
 
 
 ,
 
 
 {\displaystyle \mathbb {K} ^{n},}
 
 induces a <a href="/facts/Seminorm/vMuescKp">seminorm</a> 
 
 
 
 
 |
 
 f
 
 |
 
 :
 X
 →
 
 R
 
 
 
 {\displaystyle |f|:X\to \mathbb {R} }
 
 defined by 
 
 
 
 
 |
 
 f
 
 |
 
 (
 x
 )
 =
 
 |
 
 f
 (
 x
 )
 
 |
 
 
 
 {\displaystyle |f|(x)=|f(x)|}
 
 where 
 
 
 
 ker
 ⁡
 f
 =
 ker
 ⁡
 
 |
 
 f
 
 |
 
 .
 
 
 {\displaystyle \ker f=\ker |f|.}
 
 Every seminorm induces a (<a href="/facts/Metrizable_TVS/6FUHfx6W">pseudometrizable</a> <a href="/facts/Locally_convex/AGrPrayC">locally convex</a>) vector topology on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and seminorms with distinct kernels induce distinct topologies so that in particular, seminorms on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 that are induced by linear functionals with distinct kernels will induce distinct vector topologies on 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 
</li>
<li>However, while there are infinitely many vector topologies on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 when 
 
 
 
 dim
 ⁡
 X
 ≥
 2
 ,
 
 
 {\displaystyle \dim X\geq 2,}
 
 there are, up to TVS-isomorphism, only 
 
 
 
 1
 +
 dim
 ⁡
 X
 
 
 {\displaystyle 1+\dim X}
 
 vector topologies on 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 
 For instance, if 
 
 
 
 n
 :=
 dim
 ⁡
 X
 =
 2
 
 
 {\displaystyle n:=\dim X=2}
 
 then the vector topologies on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 consist of the trivial topology, the Hausdorff Euclidean topology, and then the infinitely many remaining non-trivial non-Euclidean vector topologies on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 are all TVS-isomorphic to one another.</li></ul></li></ul>
<h3>Non-vector topologies</h3>
Discrete and cofinite topologies
If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a non-trivial vector space (that is, of non-zero dimension) then the <a href="/facts/Discrete_topology/hpvXA23u">discrete topology</a> on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 (which is always <a href="/facts/Metrizable_space/BCLdo5Jf">metrizable</a>) is not a TVS topology because despite making addition and negation continuous (which makes it into a <a href="/facts/Topological_group/cAgNMMRU">topological group</a> under addition), it fails to make scalar multiplication continuous. The <a href="/facts/Cofinite_topology/LvpB23pu">cofinite topology</a> on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 (where a subset is open if and only if its complement is finite) is also not a TVS topology on 
 
 
 
 X
 .
 
 
 {\displaystyle X.}

<h2 id="linear-maps">Linear maps</h2>
A linear operator between two topological vector spaces which is continuous at one point is continuous on the whole domain. Moreover, a linear operator 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is continuous if 
 
 
 
 f
 (
 X
 )
 
 
 {\displaystyle f(X)}
 
 is bounded (as defined below) for some neighborhood 
 
 
 
 X
 
 
 {\displaystyle X}
 
 of the origin.
A <a href="/facts/Hyperplane/2s2ycYYd">hyperplane</a> in a topological vector space 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is either dense or closed. A <a href="/facts/Linear_functional/oGh7Asrz">linear functional</a> 
 
 
 
 f
 
 
 {\displaystyle f}
 
 on a topological vector space 
 
 
 
 X
 
 
 {\displaystyle X}
 
 has either dense or closed kernel. Moreover, 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is continuous if and only if its <a href="/facts/Kernel_(algebra)/GGSpUPMJ">kernel</a> is <a href="/facts/Closed_set/upYnRTUj">closed</a>.

<h2 id="types">Types</h2>
Depending on the application additional constraints are usually enforced on the topological structure of the space. In fact, several principal results in functional analysis fail to hold in general for topological vector spaces: the <a href="/facts/Closed_graph_theorem/CK2ra00W">closed graph theorem</a>, the <a href="/facts/Open_mapping_theorem_(functional_analysis)/nFChD5j5">open mapping theorem</a>, and the fact that the dual space of the space separates points in the space.
Below are some common topological vector spaces, roughly in order of increasing "niceness."

<ul><li><a href="/facts/F-space/NBCcYfcU">F-spaces</a> are <a href="/facts/Complete_space/lsckmU8G">complete</a> topological vector spaces with a translation-invariant metric.<a class="footnote-ref" id="fnref:49" href="#fn:49">49</a> These include <a href="/facts/Lp_space/gbad0Rv1">
 
 
 
 
 L
 
 p
 
 
 
 
 {\displaystyle L^{p}}
 
 spaces</a> for all 
 
 
 
 p
 >
 0.
 
 
 {\displaystyle p>0.}
 
</li>
<li><a href="/facts/Locally_convex_topological_vector_space/AGrPrayC">Locally convex topological vector spaces</a>: here each point has a <a href="/facts/Local_base/fMRM8Xht">local base</a> consisting of <a href="/facts/Convex_set/vdAuJRJl">convex sets</a>.<a class="footnote-ref" id="fnref:50" href="#fn:50">50</a> By a technique known as <a href="/facts/Minkowski_functional/1qoFAjRs">Minkowski functionals</a> it can be shown that a space is locally convex if and only if its topology can be defined by a family of seminorms.<a class="footnote-ref" id="fnref:51" href="#fn:51">51</a> Local convexity is the minimum requirement for "geometrical" arguments like the <a href="/facts/Hahn%25E2%2580%2593Banach_theorem/3AUHusvn">Hahn–Banach theorem</a>. The 
 
 
 
 
 L
 
 p
 
 
 
 
 {\displaystyle L^{p}}
 
 spaces are locally convex (in fact, Banach spaces) for all 
 
 
 
 p
 ≥
 1
 ,
 
 
 {\displaystyle p\geq 1,}
 
 but not for 
 
 
 
 0
 
<li><a href="/facts/Barrelled_space/r2H58TN3">Barrelled spaces</a>: locally convex spaces where the <a href="/facts/Banach%25E2%2580%2593Steinhaus_theorem/rbbdebsr">Banach–Steinhaus theorem</a> holds.</li>
<li><a href="/facts/Bornological_space/kpl5B2Ez">Bornological space</a>: a locally convex space where the <a href="/facts/Continuous_linear_operator/KcqjDBrJ">continuous linear operators</a> to any locally convex space are exactly the <a href="/facts/Bounded_linear_operator/LYmDTIqR">bounded linear operators</a>.</li>
<li><a href="/facts/Stereotype_space/g7Cjbw0S">Stereotype space</a>: a locally convex space satisfying a variant of <a href="/facts/Reflexive_space/g7Cjbw0S">reflexivity condition</a>, where the dual space is endowed with the topology of uniform convergence on <a href="/facts/Totally_bounded_space/H8GSU1TT">totally bounded sets</a>.</li>
<li><a href="/facts/Montel_space/HLXqXHzs">Montel space</a>: a barrelled space where every <a href="/facts/Closed_set/upYnRTUj">closed</a> and <a href="/facts/Bounded_set_(topological_vector_space)/bTsWeU9g">bounded set</a> is <a href="/facts/Compact_set/d0cgXJH7">compact</a></li>
<li><a href="/facts/Fr%25C3%25A9chet_space/9gXkQu2u">Fréchet spaces</a>: these are complete locally convex spaces where the topology comes from a translation-invariant metric, or equivalently: from a countable family of seminorms. Many interesting spaces of functions fall into this class -- 
 
 
 
 
 C
 
 ∞
 
 
 (
 
 R
 
 )
 
 
 {\displaystyle C^{\infty }(\mathbb {R} )}
 
 is a Fréchet space under the seminorms 
 
 
 
 ‖
 f
 
 ‖
 
 k
 ,
 ℓ
 
 
 =
 
 sup
 
 x
 ∈
 [
 −
 k
 ,
 k
 ]
 
 
 
 |
 
 
 f
 
 (
 ℓ
 )
 
 
 (
 x
 )
 
 |
 
 .
 
 
 {\textstyle \|f\|_{k,\ell }=\sup _{x\in [-k,k]}|f^{(\ell )}(x)|.}
 
 A locally convex F-space is a Fréchet space.<a class="footnote-ref" id="fnref:52" href="#fn:52">52</a></li>
<li><a href="/facts/LF-space/voQHEeGI">LF-spaces</a> are <a href="/facts/Limit_(category_theory)/JoLugLs3">limits</a> of <a href="/facts/Fr%25C3%25A9chet_space/9gXkQu2u">Fréchet spaces</a>. ILH spaces are <a href="/facts/Inverse_limit/f6QecRUu">inverse limits</a> of Hilbert spaces.</li>
<li><a href="/facts/Nuclear_space/4FLPWdfp">Nuclear spaces</a>: these are locally convex spaces with the property that every bounded map from the nuclear space to an arbitrary Banach space is a <a href="/facts/Nuclear_operator/kxnx5bIJ">nuclear operator</a>.</li>
<li><a href="/facts/Normed_space/rmDljfyE">Normed spaces</a> and <a href="/facts/Seminormed_space/vMuescKp">seminormed spaces</a>: locally convex spaces where the topology can be described by a single <a href="/facts/Norm_(mathematics)/xIbR4uE1">norm</a> or seminorm. In normed spaces a linear operator is continuous if and only if it is bounded.</li>
<li><a href="/facts/Banach_space/sDEIk7EC">Banach spaces</a>: Complete <a href="/facts/Normed_vector_space/rmDljfyE">normed vector spaces</a>. Most of functional analysis is formulated for Banach spaces. This class includes the 
 
 
 
 
 L
 
 p
 
 
 
 
 {\displaystyle L^{p}}
 
 spaces with 
 
 
 
 1
 ≤
 p
 ≤
 ∞
 ,
 
 
 {\displaystyle 1\leq p\leq \infty ,}
 
 the space 
 
 
 
 B
 V
 
 
 {\displaystyle BV}
 
 of <a href="/facts/Bounded_variation/tIjQTpjh">functions of bounded variation</a>, and <a href="/facts/Ba_space/CUXe79cw">certain spaces</a> of measures.</li>
<li><a href="/facts/Reflexive_space/g7Cjbw0S">Reflexive Banach spaces</a>: Banach spaces naturally isomorphic to their double dual (see below), which ensures that some geometrical arguments can be carried out. An important example which is not reflexive is <a href="/facts/Lp_space/gbad0Rv1">
 
 
 
 
 L
 
 1
 
 
 
 
 {\displaystyle L^{1}}
 
</a>, whose dual is 
 
 
 
 
 L
 
 ∞
 
 
 
 
 {\displaystyle L^{\infty }}
 
 but is strictly contained in the dual of 
 
 
 
 
 L
 
 ∞
 
 
 .
 
 
 {\displaystyle L^{\infty }.}
 
</li>
<li><a href="/facts/Hilbert_space/aDFMh4lV">Hilbert spaces</a>: these have an <a href="/facts/Inner_product/HoyjElEy">inner product</a>; even though these spaces may be infinite-dimensional, most geometrical reasoning familiar from finite dimensions can be carried out in them. These include 
 
 
 
 
 L
 
 2
 
 
 
 
 {\displaystyle L^{2}}
 
 spaces, the 
 
 
 
 
 L
 
 2
 
 
 
 
 {\displaystyle L^{2}}
 
 <a href="/facts/Sobolev_space/34qxWCyr">Sobolev spaces</a> 
 
 
 
 
 W
 
 2
 ,
 k
 
 
 ,
 
 
 {\displaystyle W^{2,k},}
 
 and <a href="/facts/Hardy_space/aN45dgx7">Hardy spaces</a>.</li>
<li><a href="/facts/Euclidean_space/R2UbzmzM">Euclidean spaces</a>: 
 
 
 
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {R} ^{n}}
 
 or 
 
 
 
 
 
 C
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {C} ^{n}}
 
 with the topology induced by the standard inner product. As pointed out in the preceding section, for a given finite 
 
 
 
 n
 ,
 
 
 {\displaystyle n,}
 
 there is only one 
 
 
 
 n
 
 
 {\displaystyle n}
 
-dimensional topological vector space, up to isomorphism. It follows from this that any finite-dimensional subspace of a TVS is closed. A characterization of finite dimensionality is that a Hausdorff TVS is locally compact if and only if it is finite-dimensional (therefore isomorphic to some Euclidean space).</li></ul>
<h2 id="dual-space">Dual space</h2>
Main articles: <a href="/facts/Algebraic_dual_space/4Uw4Knz1">Algebraic dual space</a>, <a href="/facts/Continuous_dual_space/4Uw4Knz1">Continuous dual space</a>, and <a href="/facts/Strong_dual_space/4RI1SYl6">Strong dual space</a>
Every topological vector space has a <a href="/facts/Continuous_dual_space/4Uw4Knz1">continuous dual space</a>—the set 
 
 
 
 
 X
 ′
 
 
 
 {\displaystyle X'}
 
 of all continuous linear functionals, that is, <a href="/facts/Continuous_linear_map/KcqjDBrJ">continuous linear maps</a> from the space into the base field 
 
 
 
 
 K
 
 .
 
 
 {\displaystyle \mathbb {K} .}
 
 A topology on the dual can be defined to be the coarsest topology such that the dual pairing each point evaluation 
 
 
 
 
 X
 ′
 
 →
 
 K
 
 
 
 {\displaystyle X'\to \mathbb {K} }
 
 is continuous. This turns the dual into a locally convex topological vector space. This topology is called the <a href="/facts/Weak_topology/XJASE9Up">weak-* topology</a>.<a class="footnote-ref" id="fnref:53" href="#fn:53">53</a> This may not be the only <a href="/facts/Natural_topology/FdT6dw23">natural topology</a> on the dual space; for instance, the dual of a normed space has a natural norm defined on it. However, it is very important in applications because of its compactness properties (see <a href="/facts/Banach%25E2%2580%2593Alaoglu_theorem/NILk5nDm">Banach–Alaoglu theorem</a>). Caution: Whenever 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a non-normable locally convex space, then the pairing map 
 
 
 
 
 X
 ′
 
 ×
 X
 →
 
 K
 
 
 
 {\displaystyle X'\times X\to \mathbb {K} }
 
 is never continuous, no matter which vector space topology one chooses on 
 
 
 
 
 X
 ′
 
 .
 
 
 {\displaystyle X'.}
 
 A topological vector space has a non-trivial continuous dual space if and only if it has a proper convex neighborhood of the origin.<a class="footnote-ref" id="fnref:54" href="#fn:54">54</a>

<h2 id="properties">Properties</h2>
See also: <a href="/facts/Locally_convex_topological_vector_space/AGrPrayC">Locally convex topological vector space § Properties</a>
For any 
 
 
 
 S
 ⊆
 X
 
 
 {\displaystyle S\subseteq X}
 
 of a TVS 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 the <a href="/facts/Convex_set/vdAuJRJl">convex</a> (resp. <a href="/facts/Balanced_set/Y2ar7E6g">balanced</a>, <a href="/facts/Absolutely_convex_set/rJ5HLWkg">disked</a>, closed convex, closed balanced, closed disked') hull of 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is the smallest subset of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 that has this property and contains 
 
 
 
 S
 .
 
 
 {\displaystyle S.}
 
 The closure (respectively, interior, <a href="/facts/Convex_hull/D4Id3JDS">convex hull</a>, balanced hull, disked hull) of a set 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is sometimes denoted by 
 
 
 
 
 cl
 
 X
 
 
 ⁡
 S
 
 
 {\displaystyle \operatorname {cl} _{X}S}
 
 (respectively, 
 
 
 
 
 Int
 
 X
 
 
 ⁡
 S
 ,
 
 
 {\displaystyle \operatorname {Int} _{X}S,}
 
 
 
 
 
 co
 ⁡
 S
 ,
 
 
 {\displaystyle \operatorname {co} S,}
 
 
 
 
 
 bal
 ⁡
 S
 ,
 
 
 {\displaystyle \operatorname {bal} S,}
 
 
 
 
 
 cobal
 ⁡
 S
 
 
 {\displaystyle \operatorname {cobal} S}
 
).
The <a href="/facts/Convex_hull/D4Id3JDS">convex hull</a> 
 
 
 
 co
 ⁡
 S
 
 
 {\displaystyle \operatorname {co} S}
 
 of a subset 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is equal to the set of all <a href="/facts/Convex_combination/YSDu46RI">convex combinations</a> of elements in 
 
 
 
 S
 ,
 
 
 {\displaystyle S,}
 
 which are finite <a href="/facts/Linear_combination/CVjL6kuN">linear combinations</a> of the form 
 
 
 
 
 t
 
 1
 
 
 
 s
 
 1
 
 
 +
 ⋯
 +
 
 t
 
 n
 
 
 
 s
 
 n
 
 
 
 
 {\displaystyle t_{1}s_{1}+\cdots +t_{n}s_{n}}
 
 where 
 
 
 
 n
 ≥
 1
 
 
 {\displaystyle n\geq 1}
 
 is an integer, 
 
 
 
 
 s
 
 1
 
 
 ,
 …
 ,
 
 s
 
 n
 
 
 ∈
 S
 
 
 {\displaystyle s_{1},\ldots ,s_{n}\in S}
 
 and 
 
 
 
 
 t
 
 1
 
 
 ,
 …
 ,
 
 t
 
 n
 
 
 ∈
 [
 0
 ,
 1
 ]
 
 
 {\displaystyle t_{1},\ldots ,t_{n}\in [0,1]}
 
 sum to 
 
 
 
 1.
 
 
 {\displaystyle 1.}
 
<a class="footnote-ref" id="fnref:55" href="#fn:55">55</a> The intersection of any family of convex sets is convex and the convex hull of a subset is equal to the intersection of all convex sets that contain it.<a class="footnote-ref" id="fnref:56" href="#fn:56">56</a>

<h3>Neighborhoods and open sets</h3>
Properties of neighborhoods and open sets
Every TVS is <a href="/facts/Connected_space/5CUTxfoA">connected</a><a class="footnote-ref" id="fnref:57" href="#fn:57">57</a> and <a href="/facts/Locally_connected_space/9qhzYyT7">locally connected</a><a class="footnote-ref" id="fnref:58" href="#fn:58">58</a> and any connected open subset of a TVS is <a href="/facts/Arcwise_connected/5CUTxfoA">arcwise connected</a>. If 
 
 
 
 S
 ⊆
 X
 
 
 {\displaystyle S\subseteq X}
 
 and 
 
 
 
 U
 
 
 {\displaystyle U}
 
 is an open subset of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 then 
 
 
 
 S
 +
 U
 
 
 {\displaystyle S+U}
 
 is an open set in 
 
 
 
 X
 
 
 {\displaystyle X}
 
<a class="footnote-ref" id="fnref:59" href="#fn:59">59</a> and if 
 
 
 
 S
 ⊆
 X
 
 
 {\displaystyle S\subseteq X}
 
 has non-empty interior then 
 
 
 
 S
 −
 S
 
 
 {\displaystyle S-S}
 
 is a neighborhood of the origin.<a class="footnote-ref" id="fnref:60" href="#fn:60">60</a>
The open convex subsets of a TVS 
 
 
 
 X
 
 
 {\displaystyle X}
 
 (not necessarily Hausdorff or locally convex) are exactly those that are of the form 
 
 
 
 z
 +
 {
 x
 ∈
 X
 :
 p
 (
 x
 )
 <
 1
 }
  
 =
  
 {
 x
 ∈
 X
 :
 p
 (
 x
 −
 z
 )
 <
 1
 }
 
 
 {\displaystyle z+\{x\in X:p(x)<1\}~=~\{x\in X:p(x-z)<1\}}
 
 for some 
 
 
 
 z
 ∈
 X
 
 
 {\displaystyle z\in X}
 
 and some positive continuous <a href="/facts/Sublinear_functional/u1NglYpF">sublinear functional</a> 
 
 
 
 p
 
 
 {\displaystyle p}
 
 on 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 
<a class="footnote-ref" id="fnref:61" href="#fn:61">61</a>
If 
 
 
 
 K
 
 
 {\displaystyle K}
 
 is an <a href="/facts/Absorbing_set/GCtjsUuX">absorbing</a> <a href="/facts/Absolutely_convex_set/rJ5HLWkg">disk</a> in a TVS 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and if 
 
 
 
 p
 :=
 
 p
 
 K
 
 
 
 
 {\displaystyle p:=p_{K}}
 
 is the <a href="/facts/Minkowski_functional/1qoFAjRs">Minkowski functional</a> of 
 
 
 
 K
 
 
 {\displaystyle K}
 
 then<a class="footnote-ref" id="fnref:62" href="#fn:62">62</a> 
 
 
 
 
 Int
 
 X
 
 
 ⁡
 K
  
 ⊆
  
 {
 x
 ∈
 X
 :
 p
 (
 x
 )
 <
 1
 }
  
 ⊆
  
 K
  
 ⊆
  
 {
 x
 ∈
 X
 :
 p
 (
 x
 )
 ≤
 1
 }
  
 ⊆
  
 
 cl
 
 X
 
 
 ⁡
 K
 
 
 {\displaystyle \operatorname {Int} _{X}K~\subseteq ~\{x\in X:p(x)<1\}~\subseteq ~K~\subseteq ~\{x\in X:p(x)\leq 1\}~\subseteq ~\operatorname {cl} _{X}K}
 
 where importantly, it was not assumed that 
 
 
 
 K
 
 
 {\displaystyle K}
 
 had any topological properties nor that 
 
 
 
 p
 
 
 {\displaystyle p}
 
 was continuous (which happens if and only if 
 
 
 
 K
 
 
 {\displaystyle K}
 
 is a neighborhood of the origin).
Let 
 
 
 
 τ
 
 
 {\displaystyle \tau }
 
 and 
 
 
 
 ν
 
 
 {\displaystyle \nu }
 
 be two vector topologies on 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 
 Then 
 
 
 
 τ
 ⊆
 ν
 
 
 {\displaystyle \tau \subseteq \nu }
 
 if and only if whenever a net 
 
 
 
 
 x
 
 ∙
 
 
 =
 
 
 (
 
 x
 
 i
 
 
 )
 
 
 i
 ∈
 I
 
 
 
 
 {\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}
 
 in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 converges 
 
 
 
 0
 
 
 {\displaystyle 0}
 
 in 
 
 
 
 (
 X
 ,
 ν
 )
 
 
 {\displaystyle (X,\nu )}
 
 then 
 
 
 
 
 x
 
 ∙
 
 
 →
 0
 
 
 {\displaystyle x_{\bullet }\to 0}
 
 in 
 
 
 
 (
 X
 ,
 τ
 )
 .
 
 
 {\displaystyle (X,\tau ).}
 
<a class="footnote-ref" id="fnref:63" href="#fn:63">63</a>
Let 
 
 
 
 
 
 N
 
 
 
 
 {\displaystyle {\mathcal {N}}}
 
 be a neighborhood basis of the origin in 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 let 
 
 
 
 S
 ⊆
 X
 ,
 
 
 {\displaystyle S\subseteq X,}
 
 and let 
 
 
 
 x
 ∈
 X
 .
 
 
 {\displaystyle x\in X.}
 
 Then 
 
 
 
 x
 ∈
 
 cl
 
 X
 
 
 ⁡
 S
 
 
 {\displaystyle x\in \operatorname {cl} _{X}S}
 
 if and only if there exists a net 
 
 
 
 
 s
 
 ∙
 
 
 =
 
 
 (
 
 s
 
 N
 
 
 )
 
 
 N
 ∈
 
 
 N
 
 
 
 
 
 
 {\displaystyle s_{\bullet }=\left(s_{N}\right)_{N\in {\mathcal {N}}}}
 
 in 
 
 
 
 S
 
 
 {\displaystyle S}
 
 (indexed by 
 
 
 
 
 
 N
 
 
 
 
 {\displaystyle {\mathcal {N}}}
 
) such that 
 
 
 
 
 s
 
 ∙
 
 
 →
 x
 
 
 {\displaystyle s_{\bullet }\to x}
 
 in 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 
<a class="footnote-ref" id="fnref:64" href="#fn:64">64</a> This shows, in particular, that it will often suffice to consider nets indexed by a neighborhood basis of the origin rather than nets on arbitrary directed sets.
If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a TVS that is of the <a href="/facts/Second_category/AxTk8frw">second category</a> in itself (that is, a <a href="/facts/Nonmeager_space/AxTk8frw">nonmeager space</a>) then any closed convex <a href="/facts/Absorbing_set/GCtjsUuX">absorbing</a> subset of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a neighborhood of the origin.<a class="footnote-ref" id="fnref:65" href="#fn:65">65</a> This is no longer guaranteed if the set is not convex (a counter-example exists even in 
 
 
 
 X
 =
 
 
 R
 
 
 2
 
 
 
 
 {\displaystyle X=\mathbb {R} ^{2}}
 
) or if 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is not of the second category in itself.<a class="footnote-ref" id="fnref:66" href="#fn:66">66</a>
Interior
If 
 
 
 
 R
 ,
 S
 ⊆
 X
 
 
 {\displaystyle R,S\subseteq X}
 
 and 
 
 
 
 S
 
 
 {\displaystyle S}
 
 has non-empty interior then

Int
          
            X
          
        
        ⁡
        S
         
        =
         
        
          Int
          
            X
          
        
        ⁡
        
          (
          
            
              cl
              
                X
              
            
            ⁡
            S
          
          )
        
         
        
           and 
        
         
        
          cl
          
            X
          
        
        ⁡
        S
         
        =
         
        
          cl
          
            X
          
        
        ⁡
        
          (
          
            
              Int
              
                X
              
            
            ⁡
            S
          
          )
        
      
    
    {\displaystyle \operatorname {Int} _{X}S~=~\operatorname {Int} _{X}\left(\operatorname {cl} _{X}S\right)~{\text{ and }}~\operatorname {cl} _{X}S~=~\operatorname {cl} _{X}\left(\operatorname {Int} _{X}S\right)}

and

Int
          
            X
          
        
        ⁡
        (
        R
        )
        +
        
          Int
          
            X
          
        
        ⁡
        (
        S
        )
         
        ⊆
         
        R
        +
        
          Int
          
            X
          
        
        ⁡
        S
        ⊆
        
          Int
          
            X
          
        
        ⁡
        (
        R
        +
        S
        )
        .
      
    
    {\displaystyle \operatorname {Int} _{X}(R)+\operatorname {Int} _{X}(S)~\subseteq ~R+\operatorname {Int} _{X}S\subseteq \operatorname {Int} _{X}(R+S).}

The <a href="/facts/Topological_interior/5sxGnQfY">topological interior</a> of a <a href="/facts/Absolutely_convex_set/rJ5HLWkg">disk</a> is not empty if and only if this interior contains the origin.<a class="footnote-ref" id="fnref:67" href="#fn:67">67</a> 
More generally, if 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is a <a href="/facts/Balanced_set/Y2ar7E6g">balanced</a> set with non-empty interior 
 
 
 
 
 Int
 
 X
 
 
 ⁡
 S
 ≠
 ∅
 
 
 {\displaystyle \operatorname {Int} _{X}S\neq \varnothing }
 
 in a TVS 
 
 
 
 X
 
 
 {\displaystyle X}
 
 then 
 
 
 
 {
 0
 }
 ∪
 
 Int
 
 X
 
 
 ⁡
 S
 
 
 {\displaystyle \{0\}\cup \operatorname {Int} _{X}S}
 
 will necessarily be balanced;<a class="footnote-ref" id="fnref:68" href="#fn:68">68</a> consequently, 
 
 
 
 
 Int
 
 X
 
 
 ⁡
 S
 
 
 {\displaystyle \operatorname {Int} _{X}S}
 
 will be balanced if and only if it contains the origin.<a class="footnote-ref" id="fnref:69" href="#fn:69">69</a> For this (i.e. 
 
 
 
 0
 ∈
 
 Int
 
 X
 
 
 ⁡
 S
 
 
 {\displaystyle 0\in \operatorname {Int} _{X}S}
 
) to be true, it suffices for 
 
 
 
 S
 
 
 {\displaystyle S}
 
 to also be convex (in addition to being balanced and having non-empty interior).;<a class="footnote-ref" id="fnref:70" href="#fn:70">70</a> 
The conclusion 
 
 
 
 0
 ∈
 
 Int
 
 X
 
 
 ⁡
 S
 
 
 {\displaystyle 0\in \operatorname {Int} _{X}S}
 
 could be false if 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is not also convex;<a class="footnote-ref" id="fnref:71" href="#fn:71">71</a> for example, in 
 
 
 
 X
 :=
 
 
 R
 
 
 2
 
 
 ,
 
 
 {\displaystyle X:=\mathbb {R} ^{2},}
 
 the interior of the closed and balanced set 
 
 
 
 S
 :=
 {
 (
 x
 ,
 y
 )
 :
 x
 y
 ≥
 0
 }
 
 
 {\displaystyle S:=\{(x,y):xy\geq 0\}}
 
 is 
 
 
 
 {
 (
 x
 ,
 y
 )
 :
 x
 y
 >
 0
 }
 .
 
 
 {\displaystyle \{(x,y):xy>0\}.}

If 
 
 
 
 C
 
 
 {\displaystyle C}
 
 is convex and 
 
 
 
 0
 <
 t
 ≤
 1
 ,
 
 
 {\displaystyle 0<t\leq 1,}
 
 then<a class="footnote-ref" id="fnref:72" href="#fn:72">72</a> 
 
 
 
 t
 Int
 ⁡
 C
 +
 (
 1
 −
 t
 )
 cl
 ⁡
 C
  
 ⊆
  
 Int
 ⁡
 C
 .
 
 
 {\displaystyle t\operatorname {Int} C+(1-t)\operatorname {cl} C~\subseteq ~\operatorname {Int} C.}
 
 
Explicitly, this means that if 
 
 
 
 C
 
 
 {\displaystyle C}
 
 is a convex subset of a TVS 
 
 
 
 X
 
 
 {\displaystyle X}
 
 (not necessarily Hausdorff or locally convex), 
 
 
 
 y
 ∈
 
 int
 
 X
 
 
 ⁡
 C
 ,
 
 
 {\displaystyle y\in \operatorname {int} _{X}C,}
 
 and 
 
 
 
 x
 ∈
 
 cl
 
 X
 
 
 ⁡
 C
 
 
 {\displaystyle x\in \operatorname {cl} _{X}C}
 
 then the open line segment joining 
 
 
 
 x
 
 
 {\displaystyle x}
 
 and 
 
 
 
 y
 
 
 {\displaystyle y}
 
 belongs to the interior of 
 
 
 
 C
 ;
 
 
 {\displaystyle C;}
 
 that is, 
 
 
 
 {
 t
 x
 +
 (
 1
 −
 t
 )
 y
 :
 0
 <
 t
 <
 1
 }
 ⊆
 
 int
 
 X
 
 
 ⁡
 C
 .
 
 
 {\displaystyle \{tx+(1-t)y:0<t<1\}\subseteq \operatorname {int} _{X}C.}
 
<a class="footnote-ref" id="fnref:73" href="#fn:73">73</a><a class="footnote-ref" id="fnref:74" href="#fn:74">74</a><a class="footnote-ref" id="fnref:75" href="#fn:75">75</a>
If 
 
 
 
 N
 ⊆
 X
 
 
 {\displaystyle N\subseteq X}
 
 is any balanced neighborhood of the origin in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 then 
 
 
 
 
 Int
 
 X
 
 
 ⁡
 N
 ⊆
 
 B
 
 1
 
 
 N
 =
 
 ⋃
 
 0
 <
 
 |
 
 a
 
 |
 
 <
 1
 
 
 a
 N
 ⊆
 N
 
 
 {\textstyle \operatorname {Int} _{X}N\subseteq B_{1}N=\bigcup _{0<|a|<1}aN\subseteq N}
 
 where 
 
 
 
 
 B
 
 1
 
 
 
 
 {\displaystyle B_{1}}
 
 is the set of all scalars 
 
 
 
 a
 
 
 {\displaystyle a}
 
 such that 
 
 
 
 
 |
 
 a
 
 |
 
 <
 1.
 
 
 {\displaystyle |a|<1.}

If 
 
 
 
 x
 
 
 {\displaystyle x}
 
 belongs to the interior of a convex set 
 
 
 
 S
 ⊆
 X
 
 
 {\displaystyle S\subseteq X}
 
 and 
 
 
 
 y
 ∈
 
 cl
 
 X
 
 
 ⁡
 S
 ,
 
 
 {\displaystyle y\in \operatorname {cl} _{X}S,}
 
 then the half-open line segment 
 
 
 
 [
 x
 ,
 y
 )
 :=
 {
 t
 x
 +
 (
 1
 −
 t
 )
 y
 :
 0
 <
 t
 ≤
 1
 }
 ⊆
 
 Int
 
 X
 
 
 ⁡
 
  if 
 
 x
 ≠
 y
 
 
 {\displaystyle [x,y):=\{tx+(1-t)y:0<t\leq 1\}\subseteq \operatorname {Int} _{X}{\text{ if }}x\neq y}
 
 and<a class="footnote-ref" id="fnref:76" href="#fn:76">76</a> 
 
 
 
 [
 x
 ,
 x
 )
 =
 ∅
 
  if 
 
 x
 =
 y
 .
 
 
 {\displaystyle [x,x)=\varnothing {\text{ if }}x=y.}
 
 
If 
 
 
 
 N
 
 
 {\displaystyle N}
 
 is a <a href="/facts/Balanced_set/Y2ar7E6g">balanced</a> neighborhood of 
 
 
 
 0
 
 
 {\displaystyle 0}
 
 in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 
 B
 
 1
 
 
 :=
 {
 a
 ∈
 
 K
 
 :
 
 |
 
 a
 
 |
 
 <
 1
 }
 ,
 
 
 {\displaystyle B_{1}:=\{a\in \mathbb {K} :|a|<1\},}
 
 then by considering intersections of the form 
 
 
 
 N
 ∩
 
 R
 
 x
 
 
 {\displaystyle N\cap \mathbb {R} x}
 
 (which are convex <a href="/facts/Symmetric_set/npzCTDRx">symmetric</a> neighborhoods of 
 
 
 
 0
 
 
 {\displaystyle 0}
 
 in the real TVS 
 
 
 
 
 R
 
 x
 
 
 {\displaystyle \mathbb {R} x}
 
) it follows that: 
 
 
 
 Int
 ⁡
 N
 =
 [
 0
 ,
 1
 )
 Int
 ⁡
 N
 =
 (
 −
 1
 ,
 1
 )
 N
 =
 
 B
 
 1
 
 
 N
 ,
 
 
 {\displaystyle \operatorname {Int} N=[0,1)\operatorname {Int} N=(-1,1)N=B_{1}N,}
 
 and furthermore, if 
 
 
 
 x
 ∈
 Int
 ⁡
 N
 
  and 
 
 r
 :=
 sup
 {
 r
 >
 0
 :
 [
 0
 ,
 r
 )
 x
 ⊆
 N
 }
 
 
 {\displaystyle x\in \operatorname {Int} N{\text{ and }}r:=\sup\{r>0:[0,r)x\subseteq N\}}
 
 then 
 
 
 
 r
 >
 1
 
  and 
 
 [
 0
 ,
 r
 )
 x
 ⊆
 Int
 ⁡
 N
 ,
 
 
 {\displaystyle r>1{\text{ and }}[0,r)x\subseteq \operatorname {Int} N,}
 
 and if 
 
 
 
 r
 ≠
 ∞
 
 
 {\displaystyle r\neq \infty }
 
 then 
 
 
 
 r
 x
 ∈
 cl
 ⁡
 N
 ∖
 Int
 ⁡
 N
 .
 
 
 {\displaystyle rx\in \operatorname {cl} N\setminus \operatorname {Int} N.}

<h3>Non-Hausdorff spaces and the closure of the origin</h3>
A topological vector space 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is Hausdorff if and only if 
 
 
 
 {
 0
 }
 
 
 {\displaystyle \{0\}}
 
 is a closed subset of 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 or equivalently, if and only if 
 
 
 
 {
 0
 }
 =
 
 cl
 
 X
 
 
 ⁡
 {
 0
 }
 .
 
 
 {\displaystyle \{0\}=\operatorname {cl} _{X}\{0\}.}
 
 Because 
 
 
 
 {
 0
 }
 
 
 {\displaystyle \{0\}}
 
 is a vector subspace of 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 the same is true of its closure 
 
 
 
 
 cl
 
 X
 
 
 ⁡
 {
 0
 }
 ,
 
 
 {\displaystyle \operatorname {cl} _{X}\{0\},}
 
 which is referred to as the closure of the origin in 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 
 This vector space satisfies 
 
 
 
 
 cl
 
 X
 
 
 ⁡
 {
 0
 }
 =
 
 ⋂
 
 N
 ∈
 
 
 N
 
 
 (
 0
 )
 
 
 N
 
 
 {\displaystyle \operatorname {cl} _{X}\{0\}=\bigcap _{N\in {\mathcal {N}}(0)}N}
 
 so that in particular, every neighborhood of the origin in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 contains the vector space 
 
 
 
 
 cl
 
 X
 
 
 ⁡
 {
 0
 }
 
 
 {\displaystyle \operatorname {cl} _{X}\{0\}}
 
 as a subset. 
The <a href="/facts/Subspace_topology/NjU6z4Ca">subspace topology</a> on 
 
 
 
 
 cl
 
 X
 
 
 ⁡
 {
 0
 }
 
 
 {\displaystyle \operatorname {cl} _{X}\{0\}}
 
 is always the <a href="/facts/Trivial_topology/uA7RlGth">trivial topology</a>, which in particular implies that the topological vector space 
 
 
 
 
 cl
 
 X
 
 
 ⁡
 {
 0
 }
 
 
 {\displaystyle \operatorname {cl} _{X}\{0\}}
 
 a <a href="/facts/Compact_space/d0cgXJH7">compact space</a> (even if its dimension is non-zero or even infinite) and consequently also a <a href="/facts/Bounded_set_(topological_vector_space)/bTsWeU9g">bounded subset</a> of 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 
 In fact, a vector subspace of a TVS is bounded if and only if it is contained in the closure of 
 
 
 
 {
 0
 }
 .
 
 
 {\displaystyle \{0\}.}
 
<a class="footnote-ref" id="fnref:77" href="#fn:77">77</a> 
Every subset of 
 
 
 
 
 cl
 
 X
 
 
 ⁡
 {
 0
 }
 
 
 {\displaystyle \operatorname {cl} _{X}\{0\}}
 
 also carries the trivial topology and so is itself a compact, and thus also complete, <a href="/facts/Topological_subspace/NjU6z4Ca">subspace</a> (see footnote for a proof).<a class="footnote-ref" id="fnref:78" href="#fn:78">78</a> In particular, if 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is not Hausdorff then there exist subsets that are both compact and complete but not closed in 
 
 
 
 X
 
 
 {\displaystyle X}
 
;<a class="footnote-ref" id="fnref:79" href="#fn:79">79</a> for instance, this will be true of any non-empty proper subset of 
 
 
 
 
 cl
 
 X
 
 
 ⁡
 {
 0
 }
 .
 
 
 {\displaystyle \operatorname {cl} _{X}\{0\}.}

If 
 
 
 
 S
 ⊆
 X
 
 
 {\displaystyle S\subseteq X}
 
 is compact, then 
 
 
 
 
 cl
 
 X
 
 
 ⁡
 S
 =
 S
 +
 
 cl
 
 X
 
 
 ⁡
 {
 0
 }
 
 
 {\displaystyle \operatorname {cl} _{X}S=S+\operatorname {cl} _{X}\{0\}}
 
 and this set is compact. Thus the closure of a compact subset of a TVS is compact (said differently, all compact sets are <a href="/facts/Relatively_compact/z5uIqMDp">relatively compact</a>),<a class="footnote-ref" id="fnref:80" href="#fn:80">80</a> which is not guaranteed for arbitrary non-Hausdorff <a href="/facts/Topological_space/v2ut7CbK">topological spaces</a>.<a class="footnote-ref" id="fnref:81" href="#fn:81">81</a>
For every subset 
 
 
 
 S
 ⊆
 X
 ,
 
 
 {\displaystyle S\subseteq X,}
 
 
 
 
 
 S
 +
 
 cl
 
 X
 
 
 ⁡
 {
 0
 }
 ⊆
 
 cl
 
 X
 
 
 ⁡
 S
 
 
 {\displaystyle S+\operatorname {cl} _{X}\{0\}\subseteq \operatorname {cl} _{X}S}
 
 and consequently, if 
 
 
 
 S
 ⊆
 X
 
 
 {\displaystyle S\subseteq X}
 
 is open or closed in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 then 
 
 
 
 S
 +
 
 cl
 
 X
 
 
 ⁡
 {
 0
 }
 =
 S
 
 
 {\displaystyle S+\operatorname {cl} _{X}\{0\}=S}
 
<a class="footnote-ref" id="fnref:82" href="#fn:82">82</a> (so that this arbitrary open or closed subsets 
 
 
 
 S
 
 
 {\displaystyle S}
 
 can be described as a <a href="/facts/Tube_lemma/q7XbEvNJ">"tube"</a> whose vertical side is the vector space 
 
 
 
 
 cl
 
 X
 
 
 ⁡
 {
 0
 }
 
 
 {\displaystyle \operatorname {cl} _{X}\{0\}}
 
). 
For any subset 
 
 
 
 S
 ⊆
 X
 
 
 {\displaystyle S\subseteq X}
 
 of this TVS 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 the following are equivalent:

<ul><li>
 
 
 
 S
 
 
 {\displaystyle S}
 
 is <a href="/facts/Totally_bounded_space/H8GSU1TT">totally bounded</a>.</li>
<li>
 
 
 
 S
 +
 
 cl
 
 X
 
 
 ⁡
 {
 0
 }
 
 
 {\displaystyle S+\operatorname {cl} _{X}\{0\}}
 
 is totally bounded.<a class="footnote-ref" id="fnref:83" href="#fn:83">83</a></li>
<li>
 
 
 
 
 cl
 
 X
 
 
 ⁡
 S
 
 
 {\displaystyle \operatorname {cl} _{X}S}
 
 is totally bounded.<a class="footnote-ref" id="fnref:84" href="#fn:84">84</a><a class="footnote-ref" id="fnref:85" href="#fn:85">85</a></li>
<li>The image if 
 
 
 
 S
 
 
 {\displaystyle S}
 
 under the canonical quotient map 
 
 
 
 X
 →
 X
 
 /
 
 
 cl
 
 X
 
 
 ⁡
 (
 {
 0
 }
 )
 
 
 {\displaystyle X\to X/\operatorname {cl} _{X}(\{0\})}
 
 is totally bounded.<a class="footnote-ref" id="fnref:86" href="#fn:86">86</a></li></ul>
If 
 
 
 
 M
 
 
 {\displaystyle M}
 
 is a vector subspace of a TVS 
 
 
 
 X
 
 
 {\displaystyle X}
 
 then 
 
 
 
 X
 
 /
 
 M
 
 
 {\displaystyle X/M}
 
 is Hausdorff if and only if 
 
 
 
 M
 
 
 {\displaystyle M}
 
 is closed in 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 
 
Moreover, the <a href="/facts/Quotient_map/muE0qQVC">quotient map</a> 
 
 
 
 q
 :
 X
 →
 X
 
 /
 
 
 cl
 
 X
 
 
 ⁡
 {
 0
 }
 
 
 {\displaystyle q:X\to X/\operatorname {cl} _{X}\{0\}}
 
 is always a <a href="/facts/Open_and_closed_maps/abHUKX7l">closed map</a> onto the (necessarily) Hausdorff TVS.<a class="footnote-ref" id="fnref:87" href="#fn:87">87</a>
Every vector subspace of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 that is an algebraic complement of 
 
 
 
 
 cl
 
 X
 
 
 ⁡
 {
 0
 }
 
 
 {\displaystyle \operatorname {cl} _{X}\{0\}}
 
 (that is, a vector subspace 
 
 
 
 H
 
 
 {\displaystyle H}
 
 that satisfies 
 
 
 
 {
 0
 }
 =
 H
 ∩
 
 cl
 
 X
 
 
 ⁡
 {
 0
 }
 
 
 {\displaystyle \{0\}=H\cap \operatorname {cl} _{X}\{0\}}
 
 and 
 
 
 
 X
 =
 H
 +
 
 cl
 
 X
 
 
 ⁡
 {
 0
 }
 
 
 {\displaystyle X=H+\operatorname {cl} _{X}\{0\}}
 
) is a <a href="/facts/Complemented_subspace/v8koLMPI">topological complement</a> of 
 
 
 
 
 cl
 
 X
 
 
 ⁡
 {
 0
 }
 .
 
 
 {\displaystyle \operatorname {cl} _{X}\{0\}.}
 
 
Consequently, if 
 
 
 
 H
 
 
 {\displaystyle H}
 
 is an algebraic complement of 
 
 
 
 
 cl
 
 X
 
 
 ⁡
 {
 0
 }
 
 
 {\displaystyle \operatorname {cl} _{X}\{0\}}
 
 in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 then the addition map 
 
 
 
 H
 ×
 
 cl
 
 X
 
 
 ⁡
 {
 0
 }
 →
 X
 ,
 
 
 {\displaystyle H\times \operatorname {cl} _{X}\{0\}\to X,}
 
 defined by 
 
 
 
 (
 h
 ,
 n
 )
 ↦
 h
 +
 n
 
 
 {\displaystyle (h,n)\mapsto h+n}
 
 is a TVS-isomorphism, where 
 
 
 
 H
 
 
 {\displaystyle H}
 
 is necessarily Hausdorff and 
 
 
 
 
 cl
 
 X
 
 
 ⁡
 {
 0
 }
 
 
 {\displaystyle \operatorname {cl} _{X}\{0\}}
 
 has the <a href="/facts/Indiscrete_topology/uA7RlGth">indiscrete topology</a>.<a class="footnote-ref" id="fnref:88" href="#fn:88">88</a> Moreover, if 
 
 
 
 C
 
 
 {\displaystyle C}
 
 is a Hausdorff <a href="/facts/Complete_topological_vector_space/e6qtS76j">completion</a> of 
 
 
 
 H
 
 
 {\displaystyle H}
 
 then 
 
 
 
 C
 ×
 
 cl
 
 X
 
 
 ⁡
 {
 0
 }
 
 
 {\displaystyle C\times \operatorname {cl} _{X}\{0\}}
 
 is a completion of 
 
 
 
 X
 ≅
 H
 ×
 
 cl
 
 X
 
 
 ⁡
 {
 0
 }
 .
 
 
 {\displaystyle X\cong H\times \operatorname {cl} _{X}\{0\}.}
 
<a class="footnote-ref" id="fnref:89" href="#fn:89">89</a>

<h3>Closed and compact sets</h3>
Compact and totally bounded sets
A subset of a TVS is compact if and only if it is complete and <a href="/facts/Totally_bounded_space/H8GSU1TT">totally bounded</a>.<a class="footnote-ref" id="fnref:90" href="#fn:90">90</a> Thus, in a <a href="/facts/Complete_topological_vector_space/e6qtS76j">complete topological vector space</a>, a closed and totally bounded subset is compact.<a class="footnote-ref" id="fnref:91" href="#fn:91">91</a> 
A subset 
 
 
 
 S
 
 
 {\displaystyle S}
 
 of a TVS 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is <a href="/facts/Totally_bounded_space/H8GSU1TT">totally bounded</a> if and only if 
 
 
 
 
 cl
 
 X
 
 
 ⁡
 S
 
 
 {\displaystyle \operatorname {cl} _{X}S}
 
 is totally bounded,<a class="footnote-ref" id="fnref:92" href="#fn:92">92</a><a class="footnote-ref" id="fnref:93" href="#fn:93">93</a> if and only if its image under the canonical quotient map 
 
 
 
 X
 →
 X
 
 /
 
 
 cl
 
 X
 
 
 ⁡
 (
 {
 0
 }
 )
 
 
 {\displaystyle X\to X/\operatorname {cl} _{X}(\{0\})}
 
 is totally bounded.<a class="footnote-ref" id="fnref:94" href="#fn:94">94</a>
Every relatively compact set is totally bounded<a class="footnote-ref" id="fnref:95" href="#fn:95">95</a> and the closure of a totally bounded set is totally bounded.<a class="footnote-ref" id="fnref:96" href="#fn:96">96</a> 
The image of a totally bounded set under a uniformly continuous map (such as a continuous linear map for instance) is totally bounded.<a class="footnote-ref" id="fnref:97" href="#fn:97">97</a> 
If 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is a subset of a TVS 
 
 
 
 X
 
 
 {\displaystyle X}
 
 such that every sequence in 
 
 
 
 S
 
 
 {\displaystyle S}
 
 has a cluster point in 
 
 
 
 S
 
 
 {\displaystyle S}
 
 then 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is totally bounded.<a class="footnote-ref" id="fnref:98" href="#fn:98">98</a>
If 
 
 
 
 K
 
 
 {\displaystyle K}
 
 is a compact subset of a TVS 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 U
 
 
 {\displaystyle U}
 
 is an open subset of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 containing 
 
 
 
 K
 ,
 
 
 {\displaystyle K,}
 
 then there exists a neighborhood 
 
 
 
 N
 
 
 {\displaystyle N}
 
 of 0 such that 
 
 
 
 K
 +
 N
 ⊆
 U
 .
 
 
 {\displaystyle K+N\subseteq U.}
 
<a class="footnote-ref" id="fnref:99" href="#fn:99">99</a>
Closure and closed set
The closure of any convex (respectively, any balanced, any absorbing) subset of any TVS has this same property. In particular, the closure of any convex, balanced, and absorbing subset is a <a href="/facts/Barrelled_space/r2H58TN3">barrel</a>.
The closure of a vector subspace of a TVS is a vector subspace. Every finite dimensional vector subspace of a Hausdorff TVS is closed. 
The sum of a closed vector subspace and a finite-dimensional vector subspace is closed.<a class="footnote-ref" id="fnref:100" href="#fn:100">100</a> 
If 
 
 
 
 M
 
 
 {\displaystyle M}
 
 is a vector subspace of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 N
 
 
 {\displaystyle N}
 
 is a closed neighborhood of the origin in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 such that 
 
 
 
 U
 ∩
 N
 
 
 {\displaystyle U\cap N}
 
 is closed in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 then 
 
 
 
 M
 
 
 {\displaystyle M}
 
 is closed in 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 
<a class="footnote-ref" id="fnref:101" href="#fn:101">101</a> 
The sum of a compact set and a closed set is closed. However, the sum of two closed subsets may fail to be closed<a class="footnote-ref" id="fnref:102" href="#fn:102">102</a> (see this footnote<a class="footnote-ref" id="fnref:103" href="#fn:103">103</a> for examples). 
If 
 
 
 
 S
 ⊆
 X
 
 
 {\displaystyle S\subseteq X}
 
 and 
 
 
 
 a
 
 
 {\displaystyle a}
 
 is a scalar then 
 
 
 
 a
 
 cl
 
 X
 
 
 ⁡
 S
 ⊆
 
 cl
 
 X
 
 
 ⁡
 (
 a
 S
 )
 ,
 
 
 {\displaystyle a\operatorname {cl} _{X}S\subseteq \operatorname {cl} _{X}(aS),}
 
 where if 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is Hausdorff, 
 
 
 
 a
 ≠
 0
 ,
 
  or 
 
 S
 =
 ∅
 
 
 {\displaystyle a\neq 0,{\text{ or }}S=\varnothing }
 
 then equality holds: 
 
 
 
 
 cl
 
 X
 
 
 ⁡
 (
 a
 S
 )
 =
 a
 
 cl
 
 X
 
 
 ⁡
 S
 .
 
 
 {\displaystyle \operatorname {cl} _{X}(aS)=a\operatorname {cl} _{X}S.}
 
 In particular, every non-zero scalar multiple of a closed set is closed. 
If 
 
 
 
 S
 ⊆
 X
 
 
 {\displaystyle S\subseteq X}
 
 and if 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is a set of scalars such that neither 
 
 
 
 cl
 ⁡
 S
 
  nor 
 
 cl
 ⁡
 A
 
 
 {\displaystyle \operatorname {cl} S{\text{ nor }}\operatorname {cl} A}
 
 contain zero then<a class="footnote-ref" id="fnref:104" href="#fn:104">104</a> 
 
 
 
 
 (
 
 cl
 ⁡
 A
 
 )
 
 
 (
 
 
 cl
 
 X
 
 
 ⁡
 S
 
 )
 
 =
 
 cl
 
 X
 
 
 ⁡
 (
 A
 S
 )
 .
 
 
 {\displaystyle \left(\operatorname {cl} A\right)\left(\operatorname {cl} _{X}S\right)=\operatorname {cl} _{X}(AS).}

If 
 
 
 
 S
 ⊆
 X
 
  and 
 
 S
 +
 S
 ⊆
 2
 
 cl
 
 X
 
 
 ⁡
 S
 
 
 {\displaystyle S\subseteq X{\text{ and }}S+S\subseteq 2\operatorname {cl} _{X}S}
 
 then 
 
 
 
 
 cl
 
 X
 
 
 ⁡
 S
 
 
 {\displaystyle \operatorname {cl} _{X}S}
 
 is convex.<a class="footnote-ref" id="fnref:105" href="#fn:105">105</a>
If 
 
 
 
 R
 ,
 S
 ⊆
 X
 
 
 {\displaystyle R,S\subseteq X}
 
 then<a class="footnote-ref" id="fnref:106" href="#fn:106">106</a> 
 
 
 
 
 cl
 
 X
 
 
 ⁡
 (
 R
 )
 +
 
 cl
 
 X
 
 
 ⁡
 (
 S
 )
  
 ⊆
  
 
 cl
 
 X
 
 
 ⁡
 (
 R
 +
 S
 )
  
 
  and 
 
  
 
 cl
 
 X
 
 
 ⁡
 
 [
 
 
 cl
 
 X
 
 
 ⁡
 (
 R
 )
 +
 
 cl
 
 X
 
 
 ⁡
 (
 S
 )
 
 ]
 
  
 =
  
 
 cl
 
 X
 
 
 ⁡
 (
 R
 +
 S
 )
 
 
 {\displaystyle \operatorname {cl} _{X}(R)+\operatorname {cl} _{X}(S)~\subseteq ~\operatorname {cl} _{X}(R+S)~{\text{ and }}~\operatorname {cl} _{X}\left[\operatorname {cl} _{X}(R)+\operatorname {cl} _{X}(S)\right]~=~\operatorname {cl} _{X}(R+S)}
 
 and so consequently, if 
 
 
 
 R
 +
 S
 
 
 {\displaystyle R+S}
 
 is closed then so is 
 
 
 
 
 cl
 
 X
 
 
 ⁡
 (
 R
 )
 +
 
 cl
 
 X
 
 
 ⁡
 (
 S
 )
 .
 
 
 {\displaystyle \operatorname {cl} _{X}(R)+\operatorname {cl} _{X}(S).}
 
<a class="footnote-ref" id="fnref:107" href="#fn:107">107</a>
If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a real TVS and 
 
 
 
 S
 ⊆
 X
 ,
 
 
 {\displaystyle S\subseteq X,}
 
 then 
 
 
 
 
 ⋂
 
 r
 >
 1
 
 
 r
 S
 ⊆
 
 cl
 
 X
 
 
 ⁡
 S
 
 
 {\displaystyle \bigcap _{r>1}rS\subseteq \operatorname {cl} _{X}S}
 
 where the left hand side is independent of the topology on 
 
 
 
 X
 ;
 
 
 {\displaystyle X;}
 
 moreover, if 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is a convex neighborhood of the origin then equality holds.
For any subset 
 
 
 
 S
 ⊆
 X
 ,
 
 
 {\displaystyle S\subseteq X,}
 
 
 
 
 
 
 cl
 
 X
 
 
 ⁡
 S
  
 =
  
 
 ⋂
 
 N
 ∈
 
 
 N
 
 
 
 
 (
 S
 +
 N
 )
 
 
 {\displaystyle \operatorname {cl} _{X}S~=~\bigcap _{N\in {\mathcal {N}}}(S+N)}
 
 where 
 
 
 
 
 
 N
 
 
 
 
 {\displaystyle {\mathcal {N}}}
 
 is any neighborhood basis at the origin for 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 
<a class="footnote-ref" id="fnref:108" href="#fn:108">108</a> 
However, 
 
 
 
 
 cl
 
 X
 
 
 ⁡
 U
  
 ⊇
  
 ⋂
 {
 U
 :
 S
 ⊆
 U
 ,
 U
 
  is open in 
 
 X
 }
 
 
 {\displaystyle \operatorname {cl} _{X}U~\supseteq ~\bigcap \{U:S\subseteq U,U{\text{ is open in }}X\}}
 
 and it is possible for this containment to be proper<a class="footnote-ref" id="fnref:109" href="#fn:109">109</a> (for example, if 
 
 
 
 X
 =
 
 R
 
 
 
 {\displaystyle X=\mathbb {R} }
 
 and 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is the rational numbers). It follows that 
 
 
 
 
 cl
 
 X
 
 
 ⁡
 U
 ⊆
 U
 +
 U
 
 
 {\displaystyle \operatorname {cl} _{X}U\subseteq U+U}
 
 for every neighborhood 
 
 
 
 U
 
 
 {\displaystyle U}
 
 of the origin in 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 
<a class="footnote-ref" id="fnref:110" href="#fn:110">110</a>
Closed hulls
In a locally convex space, convex hulls of bounded sets are bounded. This is not true for TVSs in general.<a class="footnote-ref" id="fnref:111" href="#fn:111">111</a>

<ul><li>The closed convex hull of a set is equal to the closure of the convex hull of that set; that is, equal to 
 
 
 
 
 cl
 
 X
 
 
 ⁡
 (
 co
 ⁡
 S
 )
 .
 
 
 {\displaystyle \operatorname {cl} _{X}(\operatorname {co} S).}
 
<a class="footnote-ref" id="fnref:112" href="#fn:112">112</a></li>
<li>The closed balanced hull of a set is equal to the closure of the balanced hull of that set; that is, equal to 
 
 
 
 
 cl
 
 X
 
 
 ⁡
 (
 bal
 ⁡
 S
 )
 .
 
 
 {\displaystyle \operatorname {cl} _{X}(\operatorname {bal} S).}
 
<a class="footnote-ref" id="fnref:113" href="#fn:113">113</a></li>
<li>The closed <a href="/facts/Absolutely_convex_set/rJ5HLWkg">disked</a> hull of a set is equal to the closure of the disked hull of that set; that is, equal to 
 
 
 
 
 cl
 
 X
 
 
 ⁡
 (
 cobal
 ⁡
 S
 )
 .
 
 
 {\displaystyle \operatorname {cl} _{X}(\operatorname {cobal} S).}
 
<a class="footnote-ref" id="fnref:114" href="#fn:114">114</a></li></ul>
If 
 
 
 
 R
 ,
 S
 ⊆
 X
 
 
 {\displaystyle R,S\subseteq X}
 
 and the closed convex hull of one of the sets 
 
 
 
 S
 
 
 {\displaystyle S}
 
 or 
 
 
 
 R
 
 
 {\displaystyle R}
 
 is compact then<a class="footnote-ref" id="fnref:115" href="#fn:115">115</a> 
 
 
 
 
 cl
 
 X
 
 
 ⁡
 (
 co
 ⁡
 (
 R
 +
 S
 )
 )
  
 =
  
 
 cl
 
 X
 
 
 ⁡
 (
 co
 ⁡
 R
 )
 +
 
 cl
 
 X
 
 
 ⁡
 (
 co
 ⁡
 S
 )
 .
 
 
 {\displaystyle \operatorname {cl} _{X}(\operatorname {co} (R+S))~=~\operatorname {cl} _{X}(\operatorname {co} R)+\operatorname {cl} _{X}(\operatorname {co} S).}
 
 
If 
 
 
 
 R
 ,
 S
 ⊆
 X
 
 
 {\displaystyle R,S\subseteq X}
 
 each have a closed convex hull that is compact (that is, 
 
 
 
 
 cl
 
 X
 
 
 ⁡
 (
 co
 ⁡
 R
 )
 
 
 {\displaystyle \operatorname {cl} _{X}(\operatorname {co} R)}
 
 and 
 
 
 
 
 cl
 
 X
 
 
 ⁡
 (
 co
 ⁡
 S
 )
 
 
 {\displaystyle \operatorname {cl} _{X}(\operatorname {co} S)}
 
 are compact) then<a class="footnote-ref" id="fnref:116" href="#fn:116">116</a> 
 
 
 
 
 cl
 
 X
 
 
 ⁡
 (
 co
 ⁡
 (
 R
 ∪
 S
 )
 )
  
 =
  
 co
 ⁡
 
 [
 
 
 cl
 
 X
 
 
 ⁡
 (
 co
 ⁡
 R
 )
 ∪
 
 cl
 
 X
 
 
 ⁡
 (
 co
 ⁡
 S
 )
 
 ]
 
 .
 
 
 {\displaystyle \operatorname {cl} _{X}(\operatorname {co} (R\cup S))~=~\operatorname {co} \left[\operatorname {cl} _{X}(\operatorname {co} R)\cup \operatorname {cl} _{X}(\operatorname {co} S)\right].}

Hulls and compactness
In a general TVS, the closed convex hull of a compact set may fail to be compact. 
The balanced hull of a compact (respectively, <a href="/facts/Totally_bounded/H8GSU1TT">totally bounded</a>) set has that same property.<a class="footnote-ref" id="fnref:117" href="#fn:117">117</a> 
The convex hull of a finite union of compact convex sets is again compact and convex.<a class="footnote-ref" id="fnref:118" href="#fn:118">118</a>

<h3>Other properties</h3>
Meager, nowhere dense, and Baire
A <a href="/facts/Absolutely_convex_set/rJ5HLWkg">disk</a> in a TVS is not <a href="/facts/Nowhere_dense/m0oF4X8R">nowhere dense</a> if and only if its closure is a neighborhood of the origin.<a class="footnote-ref" id="fnref:119" href="#fn:119">119</a> 
A vector subspace of a TVS that is closed but not open is <a href="/facts/Nowhere_dense/m0oF4X8R">nowhere dense</a>.<a class="footnote-ref" id="fnref:120" href="#fn:120">120</a>
Suppose 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a TVS that does not carry the <a href="/facts/Indiscrete_topology/uA7RlGth">indiscrete topology</a>. Then 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a <a href="/facts/Baire_space/HToekdT8">Baire space</a> if and only if 
 
 
 
 X
 
 
 {\displaystyle X}
 
 has no balanced absorbing nowhere dense subset.<a class="footnote-ref" id="fnref:121" href="#fn:121">121</a>
A TVS 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a Baire space if and only if 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is <a href="/facts/Nonmeager/AxTk8frw">nonmeager</a>, which happens if and only if there does not exist a <a href="/facts/Nowhere_dense/m0oF4X8R">nowhere dense</a> set 
 
 
 
 D
 
 
 {\displaystyle D}
 
 such that 
 
 
 
 X
 =
 
 ⋃
 
 n
 ∈
 
 N
 
 
 
 n
 D
 .
 
 
 {\textstyle X=\bigcup _{n\in \mathbb {N} }nD.}
 
<a class="footnote-ref" id="fnref:122" href="#fn:122">122</a> 
Every <a href="/facts/Nonmeager/AxTk8frw">nonmeager</a> locally convex TVS is a <a href="/facts/Barrelled_space/r2H58TN3">barrelled space</a>.<a class="footnote-ref" id="fnref:123" href="#fn:123">123</a>
Important algebraic facts and common misconceptions
If 
 
 
 
 S
 ⊆
 X
 
 
 {\displaystyle S\subseteq X}
 
 then 
 
 
 
 2
 S
 ⊆
 S
 +
 S
 
 
 {\displaystyle 2S\subseteq S+S}
 
; if 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is convex then equality holds. For an example where equality does not hold, let 
 
 
 
 x
 
 
 {\displaystyle x}
 
 be non-zero and set 
 
 
 
 S
 =
 {
 −
 x
 ,
 x
 }
 ;
 
 
 {\displaystyle S=\{-x,x\};}
 
 
 
 
 
 S
 =
 {
 x
 ,
 2
 x
 }
 
 
 {\displaystyle S=\{x,2x\}}
 
 also works.
A subset 
 
 
 
 C
 
 
 {\displaystyle C}
 
 is convex if and only if 
 
 
 
 (
 s
 +
 t
 )
 C
 =
 s
 C
 +
 t
 C
 
 
 {\displaystyle (s+t)C=sC+tC}
 
 for all positive real 
 
 
 
 s
 >
 0
 
  and 
 
 t
 >
 0
 ,
 
 
 {\displaystyle s>0{\text{ and }}t>0,}
 
<a class="footnote-ref" id="fnref:124" href="#fn:124">124</a> or equivalently, if and only if 
 
 
 
 t
 C
 +
 (
 1
 −
 t
 )
 C
 ⊆
 C
 
 
 {\displaystyle tC+(1-t)C\subseteq C}
 
 for all 
 
 
 
 0
 ≤
 t
 ≤
 1.
 
 
 {\displaystyle 0\leq t\leq 1.}
 
<a class="footnote-ref" id="fnref:125" href="#fn:125">125</a>
The <a href="/facts/Convex_balanced_hull/rJ5HLWkg">convex balanced hull</a> of a set 
 
 
 
 S
 ⊆
 X
 
 
 {\displaystyle S\subseteq X}
 
 is equal to the convex hull of the <a href="/facts/Balanced_hull/Y2ar7E6g">balanced hull</a> of 
 
 
 
 S
 ;
 
 
 {\displaystyle S;}
 
 that is, it is equal to 
 
 
 
 co
 ⁡
 (
 bal
 ⁡
 S
 )
 .
 
 
 {\displaystyle \operatorname {co} (\operatorname {bal} S).}
 
 But in general, 
 
 
 
 bal
 ⁡
 (
 co
 ⁡
 S
 )
  
 ⊆
  
 cobal
 ⁡
 S
  
 =
  
 co
 ⁡
 (
 bal
 ⁡
 S
 )
 ,
 
 
 {\displaystyle \operatorname {bal} (\operatorname {co} S)~\subseteq ~\operatorname {cobal} S~=~\operatorname {co} (\operatorname {bal} S),}
 
 where the inclusion might be strict since the <a href="/facts/Balanced_hull/Y2ar7E6g">balanced hull</a> of a convex set need not be convex (counter-examples exist even in 
 
 
 
 
 
 R
 
 
 2
 
 
 
 
 {\displaystyle \mathbb {R} ^{2}}
 
).
If 
 
 
 
 R
 ,
 S
 ⊆
 X
 
 
 {\displaystyle R,S\subseteq X}
 
 and 
 
 
 
 a
 
 
 {\displaystyle a}
 
 is a scalar then<a class="footnote-ref" id="fnref:126" href="#fn:126">126</a> 
 
 
 
 a
 (
 R
 +
 S
 )
 =
 a
 R
 +
 a
 S
 ,
  
 
  and 
 
  
 co
 ⁡
 (
 R
 +
 S
 )
 =
 co
 ⁡
 R
 +
 co
 ⁡
 S
 ,
  
 
  and 
 
  
 co
 ⁡
 (
 a
 S
 )
 =
 a
 co
 ⁡
 S
 .
 
 
 {\displaystyle a(R+S)=aR+aS,~{\text{ and }}~\operatorname {co} (R+S)=\operatorname {co} R+\operatorname {co} S,~{\text{ and }}~\operatorname {co} (aS)=a\operatorname {co} S.}
 
 
If 
 
 
 
 R
 ,
 S
 ⊆
 X
 
 
 {\displaystyle R,S\subseteq X}
 
 are convex non-empty disjoint sets and 
 
 
 
 x
 ∉
 R
 ∪
 S
 ,
 
 
 {\displaystyle x\not \in R\cup S,}
 
 then 
 
 
 
 S
 ∩
 co
 ⁡
 (
 R
 ∪
 {
 x
 }
 )
 =
 ∅
 
 
 {\displaystyle S\cap \operatorname {co} (R\cup \{x\})=\varnothing }
 
 or 
 
 
 
 R
 ∩
 co
 ⁡
 (
 S
 ∪
 {
 x
 }
 )
 =
 ∅
 .
 
 
 {\displaystyle R\cap \operatorname {co} (S\cup \{x\})=\varnothing .}

In any non-trivial vector space 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 there exist two disjoint non-empty convex subsets whose union is 
 
 
 
 X
 .
 
 
 {\displaystyle X.}

Other properties
Every TVS topology can be generated by a family of <a href="/facts/F-seminorm/6FUHfx6W">F-seminorms</a>.<a class="footnote-ref" id="fnref:127" href="#fn:127">127</a> 
If 
 
 
 
 P
 (
 x
 )
 
 
 {\displaystyle P(x)}
 
 is some unary <a href="/facts/Predicate_(mathematical_logic)/5WazAvKF">predicate</a> (a true or false statement dependent on 
 
 
 
 x
 ∈
 X
 
 
 {\displaystyle x\in X}
 
) then for any 
 
 
 
 z
 ∈
 X
 ,
 
 
 {\displaystyle z\in X,}
 
 
 
 
 
 z
 +
 {
 x
 ∈
 X
 :
 P
 (
 x
 )
 }
 =
 {
 x
 ∈
 X
 :
 P
 (
 x
 −
 z
 )
 }
 .
 
 
 {\displaystyle z+\{x\in X:P(x)\}=\{x\in X:P(x-z)\}.}
 
<a class="footnote-ref" id="fnref:128" href="#fn:128">128</a> 
So for example, if 
 
 
 
 P
 (
 x
 )
 
 
 {\displaystyle P(x)}
 
 denotes "
 
 
 
 ‖
 x
 ‖
 <
 1
 
 
 {\displaystyle \|x\|<1}
 
" then for any 
 
 
 
 z
 ∈
 X
 ,
 
 
 {\displaystyle z\in X,}
 
 
 
 
 
 z
 +
 {
 x
 ∈
 X
 :
 ‖
 x
 ‖
 <
 1
 }
 =
 {
 x
 ∈
 X
 :
 ‖
 x
 −
 z
 ‖
 <
 1
 }
 .
 
 
 {\displaystyle z+\{x\in X:\|x\|<1\}=\{x\in X:\|x-z\|<1\}.}
 
 Similarly, if 
 
 
 
 s
 ≠
 0
 
 
 {\displaystyle s\neq 0}
 
 is a scalar then 
 
 
 
 s
 {
 x
 ∈
 X
 :
 P
 (
 x
 )
 }
 =
 
 {
 
 x
 ∈
 X
 :
 P
 
 (
 
 
 
 
 1
 s
 
 
 
 x
 
 )
 
 
 }
 
 .
 
 
 {\displaystyle s\{x\in X:P(x)\}=\left\{x\in X:P\left({\tfrac {1}{s}}x\right)\right\}.}
 
 The elements 
 
 
 
 x
 ∈
 X
 
 
 {\displaystyle x\in X}
 
 of these sets must range over a vector space (that is, over 
 
 
 
 X
 
 
 {\displaystyle X}
 
) rather than not just a subset or else these equalities are no longer guaranteed; similarly, 
 
 
 
 z
 
 
 {\displaystyle z}
 
 must belong to this vector space (that is, 
 
 
 
 z
 ∈
 X
 
 
 {\displaystyle z\in X}
 
).

<h3>Properties preserved by set operators</h3>
<ul><li>The balanced hull of a compact (respectively, <a href="/facts/Totally_bounded/H8GSU1TT">totally bounded</a>, open) set has that same property.<a class="footnote-ref" id="fnref:129" href="#fn:129">129</a></li>
<li>The <a href="/facts/Minkowski_sum/dEQfoJ1A">(Minkowski) sum</a> of two compact (respectively, bounded, balanced, convex) sets has that same property.<a class="footnote-ref" id="fnref:130" href="#fn:130">130</a> But the sum of two closed sets need not be closed.</li>
<li>The convex hull of a balanced (resp. open) set is balanced (respectively, open). However, the convex hull of a closed set need not be closed.<a class="footnote-ref" id="fnref:131" href="#fn:131">131</a> And the convex hull of a bounded set need not be bounded.</li></ul>
The following table, the color of each cell indicates whether or not a given property of subsets of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 (indicated by the column name, "convex" for instance) is preserved under the set operator (indicated by the row's name, "closure" for instance). If in every TVS, a property is preserved under the indicated set operator then that cell will be colored green; otherwise, it will be colored red.
So for instance, since the union of two absorbing sets is again absorbing, the cell in row "
 
 
 
 R
 ∪
 S
 
 
 {\displaystyle R\cup S}
 
" and column "Absorbing" is colored green. But since the arbitrary intersection of absorbing sets need not be absorbing, the cell in row "Arbitrary intersections (of at least 1 set)" and column "Absorbing" is colored red. If a cell is not colored then that information has yet to be filled in.

Properties preserved by set operators<table><tbody><tr><th rowspan="2">Operation</th><th colspan="100">Property of R , {\displaystyle R,} S , {\displaystyle S,} and any other subsets of X {\displaystyle X} that is considered</th></tr><tr><th><a href="/facts/Absorbing_set/GCtjsUuX">Absorbing</a></th><th><a href="/facts/Balanced_set/Y2ar7E6g">Balanced</a></th><th><a href="/facts/Convex_set/vdAuJRJl">Convex</a></th><th><a href="/facts/Symmetric_set/npzCTDRx">Symmetric</a></th><th>ConvexBalanced</th><th>Vectorsubspace</th><th>Open</th><th>Neighborhoodof 0</th><th>Closed</th><th>ClosedBalanced</th><th>ClosedConvex</th><th>ClosedConvexBalanced</th><th><a href="/facts/Barrelled_set/5R3MydDv">Barrel</a></th><th>ClosedVectorsubspace</th><th><a href="/facts/Totally_bounded/H8GSU1TT">Totallybounded</a></th><th><a href="/facts/Compact_set/d0cgXJH7">Compact</a></th><th>CompactConvex</th><th><a href="/facts/Relatively_compact/z5uIqMDp">Relatively compact</a></th><th><a href="/facts/Complete_space/lsckmU8G">Complete</a></th><th><a href="/facts/Sequentially_complete_space/8avaV5Uu">SequentiallyComplete</a></th><th><a href="/facts/Banach_disk/JTRkjXxx">Banachdisk</a></th><th><a href="/facts/Bounded_set_(topological_vector_space)/bTsWeU9g">Bounded</a></th><th><a href="/facts/Bornivorous_set/xwVBUY0M">Bornivorous</a></th><th><a href="/facts/Infrabornivorous/xwVBUY0M">Infrabornivorous</a></th><th><a href="/facts/Nowhere_dense_set/m0oF4X8R">Nowheredense</a> (in X {\displaystyle X} )</th><th><a href="/facts/Meagre_set/AxTk8frw">Meager</a></th><th><a href="/facts/Separable_space/2bmU73bk">Separable</a></th><th><a href="/facts/Metrizable_TVS/6FUHfx6W">Pseudometrizable</a></th><th>Operation</th></tr><tr><th> R ∪ S {\displaystyle R\cup S} </th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><th> R ∪ S {\displaystyle R\cup S} </th></tr><tr><th> ∪ {\displaystyle \cup }  of increasing nonempty <a href="/facts/Chain_(order_theory)/xnTtmuYJ">chain</a></th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><th> ∪ {\displaystyle \cup }  of increasing nonempty <a href="/facts/Chain_(order_theory)/xnTtmuYJ">chain</a></th></tr><tr><th>Arbitrary unions (of at least 1 set)</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><th>Arbitrary unions (of at least 1 set)</th></tr><tr><th> R ∩ S {\displaystyle R\cap S} </th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><th> R ∩ S {\displaystyle R\cap S} </th></tr><tr><th> ∩ {\displaystyle \cap }  of decreasing nonempty <a href="/facts/Chain_(order_theory)/xnTtmuYJ">chain</a></th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><th> ∩ {\displaystyle \cap }  of decreasing nonempty <a href="/facts/Chain_(order_theory)/xnTtmuYJ">chain</a></th></tr><tr><th>Arbitrary intersections (of at least 1 set)</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><th>Arbitrary intersections (of at least 1 set)</th></tr><tr><th> R + S {\displaystyle R+S} </th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><th> R + S {\displaystyle R+S} </th></tr><tr><th>Scalar multiple</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><th>Scalar multiple</th></tr><tr><th>Non-0 scalar multiple</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><th>Non-0 scalar multiple</th></tr><tr><th>Positive scalar multiple</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><th>Positive scalar multiple</th></tr><tr><th><a href="/facts/Closure_(topology)/9x2wlgUW">Closure</a></th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><th><a href="/facts/Closure_(topology)/9x2wlgUW">Closure</a></th></tr><tr><th><a href="/facts/Interior_(topology)/5sxGnQfY">Interior</a></th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><th><a href="/facts/Interior_(topology)/5sxGnQfY">Interior</a></th></tr><tr><th><a href="/facts/Balanced_core/Y2ar7E6g">Balanced core</a></th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><th><a href="/facts/Balanced_core/Y2ar7E6g">Balanced core</a></th></tr><tr><th><a href="/facts/Balanced_hull/Y2ar7E6g">Balanced hull</a></th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><th><a href="/facts/Balanced_hull/Y2ar7E6g">Balanced hull</a></th></tr><tr><th><a href="/facts/Convex_hull/D4Id3JDS">Convex hull</a></th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><th><a href="/facts/Convex_hull/D4Id3JDS">Convex hull</a></th></tr><tr><th><a href="/facts/Convex_balanced_hull/rJ5HLWkg">Convex balanced hull</a></th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><th><a href="/facts/Convex_balanced_hull/rJ5HLWkg">Convex balanced hull</a></th></tr><tr><th>Closed balanced hull</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><th>Closed balanced hull</th></tr><tr><th>Closed convex hull</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><th>Closed convex hull</th></tr><tr><th>Closed convex balanced hull</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><th>Closed convex balanced hull</th></tr><tr><th><a href="/facts/Linear_span/vPjclEtb">Linear span</a></th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><th><a href="/facts/Linear_span/vPjclEtb">Linear span</a></th></tr><tr><th>Pre-image under a continuous linear map</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><th>Pre-image under a continuous linear map</th></tr><tr><th>Image under a continuous linear map</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><th>Image under a continuous linear map</th></tr><tr><th>Image under a continuous linear surjection</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><th>Image under a continuous linear surjection</th></tr><tr><th>Non-empty subset of R {\displaystyle R} </th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><th>Non-empty subset of R {\displaystyle R} </th></tr><tr><th>Operation</th><th><a href="/facts/Absorbing_set/GCtjsUuX">Absorbing</a></th><th><a href="/facts/Balanced_set/Y2ar7E6g">Balanced</a></th><th><a href="/facts/Convex_set/vdAuJRJl">Convex</a></th><th><a href="/facts/Symmetric_set/npzCTDRx">Symmetric</a></th><th>ConvexBalanced</th><th>Vectorsubspace</th><th>Open</th><th>Neighborhoodof 0</th><th>Closed</th><th>ClosedBalanced</th><th>ClosedConvex</th><th>ClosedConvexBalanced</th><th><a href="/facts/Barrelled_set/5R3MydDv">Barrel</a></th><th>ClosedVectorsubspace</th><th><a href="/facts/Totally_bounded/H8GSU1TT">Totallybounded</a></th><th><a href="/facts/Compact_set/d0cgXJH7">Compact</a></th><th>CompactConvex</th><th><a href="/facts/Relatively_compact/z5uIqMDp">Relatively compact</a></th><th><a href="/facts/Complete_space/lsckmU8G">Complete</a></th><th><a href="/facts/Sequentially_complete_space/8avaV5Uu">SequentiallyComplete</a></th><th><a href="/facts/Banach_disk/JTRkjXxx">Banachdisk</a></th><th><a href="/facts/Bounded_set_(topological_vector_space)/bTsWeU9g">Bounded</a></th><th><a href="/facts/Bornivorous_set/xwVBUY0M">Bornivorous</a></th><th><a href="/facts/Infrabornivorous/xwVBUY0M">Infrabornivorous</a></th><th><a href="/facts/Nowhere_dense_set/m0oF4X8R">Nowheredense</a> (in X {\displaystyle X} )</th><th><a href="/facts/Meagre_set/AxTk8frw">Meager</a></th><th><a href="/facts/Separable_space/2bmU73bk">Separable</a></th><th><a href="/facts/Metrizable_TVS/6FUHfx6W">Pseudometrizable</a></th><th>Operation</th></tr></tbody></table>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Banach_space/sDEIk7EC">Banach space</a> – Normed vector space that is complete</li>
<li><a href="/facts/Complete_field/g4H2LRRG">Complete field</a> – algebraic structure that is complete relative to a metricPages displaying wikidata descriptions as a fallback</li>
<li><a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> – Type of topological vector space</li>
<li><a href="/facts/Liquid_vector_space/pmoY5bgE">Liquid vector space</a> – Area of mathematics using condensed setsPages displaying short descriptions of redirect targets</li>
<li><a href="/facts/Normed_space/rmDljfyE">Normed space</a> – Vector space on which a distance is definedPages displaying short descriptions of redirect targets</li>
<li><a href="/facts/Locally_compact_field/rGLcSge0">Locally compact field</a></li>
<li><a href="/facts/Locally_compact_group/AGPC8vnY">Locally compact group</a> – topological group for which the underlying topology is locally compact and Hausdorff, so that the Haar measure can be definedPages displaying wikidata descriptions as a fallback</li>
<li><a href="/facts/Locally_compact_quantum_group/wPBtnXGr">Locally compact quantum group</a> – relatively new C*-algebraic approach toward quantum groupsPages displaying wikidata descriptions as a fallback</li>
<li><a href="/facts/Locally_convex_topological_vector_space/AGrPrayC">Locally convex topological vector space</a> – Vector space with a topology defined by convex open sets</li>
<li><a href="/facts/Ordered_topological_vector_space/N1fvN8th">Ordered topological vector space</a></li>
<li><a href="/facts/Topological_abelian_group/u8FSbRux">Topological abelian group</a> – topological group whose group is abelianPages displaying wikidata descriptions as a fallback</li>
<li><a href="/facts/Topological_field/qxQqzhCy">Topological field</a> – Algebraic structure with addition, multiplication, and divisionPages displaying short descriptions of redirect targets</li>
<li><a href="/facts/Topological_group/cAgNMMRU">Topological group</a> – Group that is a topological space with continuous group action</li>
<li><a href="/facts/Topological_module/AXASMgd6">Topological module</a></li>
<li><a href="/facts/Topological_ring/qxQqzhCy">Topological ring</a> – ring where ring operations are continuousPages displaying wikidata descriptions as a fallback</li>
<li><a href="/facts/Topological_semigroup/JDwmyd5M">Topological semigroup</a> – semigroup with continuous operationPages displaying wikidata descriptions as a fallback</li>
<li><a href="/facts/Topological_vector_lattice/q5BLo3cr">Topological vector lattice</a></li></ul>
<h2 id="notes">Notes</h2>

<h3>Proofs</h3>

<h2 id="citations">Citations</h2>

<h2 id="bibliography">Bibliography</h2>

<ul><li>Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. Vol. 639. Berlin New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-08662-8. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/297140003">297140003</a>.</li>
<li>Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-519-02224-4. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/8210342">8210342</a>.</li>
<li><a href="/facts/Gottfried_K%25C3%25B6the/y7A72q5g">Köthe, Gottfried</a> (1983) [1969]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-642-64988-2. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0248498">0248498</a>. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/840293704">840293704</a>.</li>
<li>Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1584888666. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/144216834">144216834</a>.</li>
<li><a href="/facts/Walter_Rudin/9mxRAECY">Rudin, Walter</a> (1991). <a href="https://archive.org/details/functionalanalys00rudi">Functional Analysis</a>. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: <a href="/facts/McGraw-Hill_Science%2fEngineering%2fMath/srSyQSsY">McGraw-Hill Science/Engineering/Math</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-07-054236-5. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/21163277">21163277</a>.</li>
<li><a href="/facts/Helmut_H._Schaefer/xorHTDBE">Schaefer, Helmut H.</a>; Wolff, Manfred P. (1999). Topological Vector Spaces. <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">GTM</a>. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4612-7155-0. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/840278135">840278135</a>.</li>
<li><a href="/facts/Eric_Schechter/cApyLyxB">Schechter, Eric</a> (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-12-622760-4. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/175294365">175294365</a>.</li>
<li>Swartz, Charles (1992). An introduction to Functional Analysis. New York: M. Dekker. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8247-8643-4. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/24909067">24909067</a>.</li>
<li><a href="/facts/Albert_Wilansky/I6QRVtGa">Wilansky, Albert</a> (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-49353-4. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/849801114">849801114</a>.</li></ul>

<h2 id="further-reading">Further reading</h2>
<ul><li>Bierstedt, Klaus-Dieter (1988). <a href="https://digital.ub.uni-paderborn.de/hsx/download/pdf/42382?originalFilename=true">"An Introduction to Locally Convex Inductive Limits"</a>. Functional Analysis and Applications. Singapore-New Jersey-Hong Kong: Universitätsbibliothek: 35–133. Retrieved 20 September 2020.</li>
<li><a href="/facts/Nicolas_Bourbaki/qRJhs9nr">Bourbaki, Nicolas</a> (1987) [1981]. Topological Vector Spaces: Chapters 1–5. <a href="/facts/%25C3%2589l%25C3%25A9ments_de_math%25C3%25A9matique/8TQroWxE">Éléments de mathématique</a>. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-13627-4. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/17499190">17499190</a>.</li>
<li><a href="/facts/John_B._Conway/7aMjnAxr">Conway, John B.</a> (1990). A Course in Functional Analysis. <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">Graduate Texts in Mathematics</a>. Vol. 96 (2nd ed.). New York: <a href="/facts/Springer_Publishing/JfRev59b">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-97245-9. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/21195908">21195908</a>.</li>
<li><a href="/facts/Nelson_Dunford/Q72PJMF7">Dunford, Nelson</a>; <a href="/facts/Jacob_T._Schwartz/UAvyPwCr">Schwartz, Jacob T.</a> (1988). <a href="/facts/Linear_Operators_(book)/RRWBcSUL">Linear Operators</a>. Pure and applied mathematics. Vol. 1. New York: <a href="/facts/Wiley-Interscience/53g3LqJc">Wiley-Interscience</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-471-60848-6. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/18412261">18412261</a>.</li>
<li>Edwards, Robert E. (1995). Functional Analysis: Theory and Applications. New York: Dover Publications. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-68143-6. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/30593138">30593138</a>.</li>
<li><a href="/facts/Alexander_Grothendieck/jLQGPY7M">Grothendieck, Alexander</a> (1973). <a href="https://archive.org/details/topologicalvecto0000grot">Topological Vector Spaces</a>. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-677-30020-7. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/886098">886098</a>.</li>
<li><a href="/facts/John_Horvath_(mathematician)/VcacePH8">Horváth, John</a> (1966). Topological Vector Spaces and Distributions. Addison-Wesley series in mathematics. Vol. 1. Reading, MA: Addison-Wesley Publishing Company. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0201029857.</li>
<li><a href="/facts/Gottfried_K%25C3%25B6the/y7A72q5g">Köthe, Gottfried</a> (1979). Topological Vector Spaces II. Grundlehren der mathematischen Wissenschaften. Vol. 237. New York: Springer Science & Business Media. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-90400-9. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/180577972">180577972</a>.</li>
<li><a href="/facts/Serge_Lang/I7MW2rUu">Lang, Serge</a> (1972). Differential manifolds. Reading, Mass.–London–Don Mills, Ont.: Addison-Wesley Publishing Co., Inc. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-201-04166-9.</li>
<li>Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. Vol. 53. Cambridge England: <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-29882-7. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/589250">589250</a>.</li>
<li><a href="/facts/Fran%25C3%25A7ois_Tr%25C3%25A8ves/cpjQbAov">Trèves, François</a> (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-45352-1. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/853623322">853623322</a>.</li>
<li>Valdivia, Manuel (1982). Nachbin, Leopoldo (ed.). Topics in Locally Convex Spaces. Vol. 67. Amsterdam New York, N.Y.: <a href="/facts/Elsevier/jkPSAEmR">Elsevier</a> Science Pub. Co. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-08-087178-3. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/316568534">316568534</a>.</li>
<li>Voigt, Jürgen (2020). A Course on Topological Vector Spaces. Compact Textbooks in Mathematics. Cham: Birkhäuser Basel. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-030-32945-7. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/1145563701">1145563701</a>.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li> Media related to Topological vector spaces at Wikimedia Commons</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Rudin 1991, p. 4-5 §1.3. - Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. <a href="https://archive.org/details/functionalanalys00rudi" target="_blank">https://archive.org/details/functionalanalys00rudi</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Köthe 1983, p. 91. - Köthe, Gottfried (1983) [1969]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0248498" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0248498</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Schaefer & Wolff 1999, pp. 74–78. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Köthe 1983, p. 91. - Köthe, Gottfried (1983) [1969]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0248498" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0248498</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Grothendieck 1973, pp. 34–36. - Grothendieck, Alexander (1973). Topological Vector Spaces. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098. <a href="https://archive.org/details/topologicalvecto0000grot" target="_blank">https://archive.org/details/topologicalvecto0000grot</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Köthe 1983, p. 91. - Köthe, Gottfried (1983) [1969]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0248498" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0248498</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Wilansky 2013, pp. 40–47. - Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114. <a href="https://search.worldcat.org/oclc/849801114" target="_blank">https://search.worldcat.org/oclc/849801114</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Wilansky 2013, pp. 40–47. - Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114. <a href="https://search.worldcat.org/oclc/849801114" target="_blank">https://search.worldcat.org/oclc/849801114</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Narici & Beckenstein 2011, pp. 67–113. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Wilansky 2013, pp. 40–47. - Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114. <a href="https://search.worldcat.org/oclc/849801114" target="_blank">https://search.worldcat.org/oclc/849801114</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Adasch, Ernst & Keim 1978, pp. 5–9. - Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. Vol. 639. Berlin New York: Springer-Verlag. ISBN 978-3-540-08662-8. OCLC 297140003. <a href="https://search.worldcat.org/oclc/297140003" target="_blank">https://search.worldcat.org/oclc/297140003</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Schechter 1996, pp. 721–751. - Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365. <a href="https://search.worldcat.org/oclc/175294365" target="_blank">https://search.worldcat.org/oclc/175294365</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Narici & Beckenstein 2011, pp. 371–423. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">The topological properties of course also require that 
 
 
 
 X
 
 
 {\displaystyle X}
 
 be a TVS. <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Adasch, Ernst & Keim 1978, pp. 5–9. - Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. Vol. 639. Berlin New York: Springer-Verlag. ISBN 978-3-540-08662-8. OCLC 297140003. <a href="https://search.worldcat.org/oclc/297140003" target="_blank">https://search.worldcat.org/oclc/297140003</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Adasch, Ernst & Keim 1978, pp. 5–9. - Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. Vol. 639. Berlin New York: Springer-Verlag. ISBN 978-3-540-08662-8. OCLC 297140003. <a href="https://search.worldcat.org/oclc/297140003" target="_blank">https://search.worldcat.org/oclc/297140003</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Adasch, Ernst & Keim 1978, pp. 5–9. - Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. Vol. 639. Berlin New York: Springer-Verlag. ISBN 978-3-540-08662-8. OCLC 297140003. <a href="https://search.worldcat.org/oclc/297140003" target="_blank">https://search.worldcat.org/oclc/297140003</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">This condition is satisfied if 
 
 
 
 
 S
 
 
 
 {\displaystyle \mathbb {S} }
 
 denotes the set of all topological strings in 
 
 
 
 (
 X
 ,
 τ
 )
 .
 
 
 {\displaystyle (X,\tau ).}
 <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Adasch, Ernst & Keim 1978, pp. 5–9. - Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. Vol. 639. Berlin New York: Springer-Verlag. ISBN 978-3-540-08662-8. OCLC 297140003. <a href="https://search.worldcat.org/oclc/297140003" target="_blank">https://search.worldcat.org/oclc/297140003</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">Adasch, Ernst & Keim 1978, pp. 10–15. - Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. Vol. 639. Berlin New York: Springer-Verlag. ISBN 978-3-540-08662-8. OCLC 297140003. <a href="https://search.worldcat.org/oclc/297140003" target="_blank">https://search.worldcat.org/oclc/297140003</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
<li id="fn:21">Wilansky 2013, p. 53. - Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114. <a href="https://search.worldcat.org/oclc/849801114" target="_blank">https://search.worldcat.org/oclc/849801114</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></li>
<li id="fn:22">In particular, 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is Hausdorff if and only if the set 
 
 
 
 {
 0
 }
 
 
 {\displaystyle \{0\}}
 
 is closed (that is, 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a T1 space). <a href="/wiki/T1_space" target="_blank">/wiki/T1_space</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></li>
<li id="fn:23">Narici & Beckenstein 2011, pp. 67–113. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></li>
<li id="fn:24">Rudin 1991, p. 6 §1.4. - Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. <a href="https://archive.org/details/functionalanalys00rudi" target="_blank">https://archive.org/details/functionalanalys00rudi</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></li>
<li id="fn:25">Rudin 1991, p. 6 §1.4. - Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. <a href="https://archive.org/details/functionalanalys00rudi" target="_blank">https://archive.org/details/functionalanalys00rudi</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></li>
<li id="fn:26">Rudin 1991, p. 6 §1.4. - Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. <a href="https://archive.org/details/functionalanalys00rudi" target="_blank">https://archive.org/details/functionalanalys00rudi</a> <a href="#fnref:26" class="footnote-back-ref">↩</a></li>
<li id="fn:27">Narici & Beckenstein 2011, pp. 67–113. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:27" class="footnote-back-ref">↩</a></li>
<li id="fn:28">Rudin 1991, p. 8. - Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. <a href="https://archive.org/details/functionalanalys00rudi" target="_blank">https://archive.org/details/functionalanalys00rudi</a> <a href="#fnref:28" class="footnote-back-ref">↩</a></li>
<li id="fn:29">Narici & Beckenstein 2011, pp. 155–176. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:29" class="footnote-back-ref">↩</a></li>
<li id="fn:30">Rudin 1991, p. 27-28 Theorem 1.37. - Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. <a href="https://archive.org/details/functionalanalys00rudi" target="_blank">https://archive.org/details/functionalanalys00rudi</a> <a href="#fnref:30" class="footnote-back-ref">↩</a></li>
<li id="fn:31">Narici & Beckenstein 2011, pp. 155–176. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:31" class="footnote-back-ref">↩</a></li>
<li id="fn:32">Narici & Beckenstein 2011, pp. 155–176. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:32" class="footnote-back-ref">↩</a></li>
<li id="fn:33">Köthe 1983, section 15.11. - Köthe, Gottfried (1983) [1969]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0248498" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0248498</a> <a href="#fnref:33" class="footnote-back-ref">↩</a></li>
<li id="fn:34">In fact, this is true for topological group, since the proof does not use the scalar multiplications. <a href="#fnref:34" class="footnote-back-ref">↩</a></li>
<li id="fn:35">Also called a metric linear space, which means that it is a real or complex vector space together with a translation-invariant metric for which addition and scalar multiplication are continuous. <a href="#fnref:35" class="footnote-back-ref">↩</a></li>
<li id="fn:36">"Topological vector space", Encyclopedia of Mathematics, EMS Press, 2001 [1994], retrieved 26 February 2021 <a href="https://www.encyclopediaofmath.org/index.php?title=Topological_vector_space" target="_blank">https://www.encyclopediaofmath.org/index.php?title=Topological_vector_space</a> <a href="#fnref:36" class="footnote-back-ref">↩</a></li>
<li id="fn:37">Rudin 1991, p. 17 Theorem 1.22. - Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. <a href="https://archive.org/details/functionalanalys00rudi" target="_blank">https://archive.org/details/functionalanalys00rudi</a> <a href="#fnref:37" class="footnote-back-ref">↩</a></li>
<li id="fn:38">Schaefer & Wolff 1999, pp. 12–19. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:38" class="footnote-back-ref">↩</a></li>
<li id="fn:39">Schaefer & Wolff 1999, p. 16. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:39" class="footnote-back-ref">↩</a></li>
<li id="fn:40">Narici & Beckenstein 2011, pp. 67–113. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:40" class="footnote-back-ref">↩</a></li>
<li id="fn:41">Narici & Beckenstein 2011, pp. 115–154. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:41" class="footnote-back-ref">↩</a></li>
<li id="fn:42">Narici & Beckenstein 2011, pp. 115–154. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:42" class="footnote-back-ref">↩</a></li>
<li id="fn:43">Narici & Beckenstein 2011, pp. 115–154. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:43" class="footnote-back-ref">↩</a></li>
<li id="fn:44">A series 
 
 
 
 
 ∑
 
 i
 =
 1
 
 
 ∞
 
 
 
 x
 
 i
 
 
 
 
 {\textstyle \sum _{i=1}^{\infty }x_{i}}
 
 is said to converge in a TVS 
 
 
 
 X
 
 
 {\displaystyle X}
 
 if the sequence of partial sums converges. <a href="#fnref:44" class="footnote-back-ref">↩</a></li>
<li id="fn:45">Swartz 1992, pp. 27–29. - Swartz, Charles (1992). An introduction to Functional Analysis. New York: M. Dekker. ISBN 978-0-8247-8643-4. OCLC 24909067. <a href="https://search.worldcat.org/oclc/24909067" target="_blank">https://search.worldcat.org/oclc/24909067</a> <a href="#fnref:45" class="footnote-back-ref">↩</a></li>
<li id="fn:46">"A quick application of the closed graph theorem". What's new. 2016-04-22. Retrieved 2020-10-07. <a href="https://terrytao.wordpress.com/2016/04/22/a-quick-application-of-the-closed-graph-theorem/" target="_blank">https://terrytao.wordpress.com/2016/04/22/a-quick-application-of-the-closed-graph-theorem/</a> <a href="#fnref:46" class="footnote-back-ref">↩</a></li>
<li id="fn:47">Narici & Beckenstein 2011, p. 111. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:47" class="footnote-back-ref">↩</a></li>
<li id="fn:48">Narici & Beckenstein 2011, p. 111. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:48" class="footnote-back-ref">↩</a></li>
<li id="fn:49">Rudin 1991, p. 9 §1.8. - Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. <a href="https://archive.org/details/functionalanalys00rudi" target="_blank">https://archive.org/details/functionalanalys00rudi</a> <a href="#fnref:49" class="footnote-back-ref">↩</a></li>
<li id="fn:50">Rudin 1991, p. 9 §1.8. - Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. <a href="https://archive.org/details/functionalanalys00rudi" target="_blank">https://archive.org/details/functionalanalys00rudi</a> <a href="#fnref:50" class="footnote-back-ref">↩</a></li>
<li id="fn:51">Rudin 1991, p. 27 Theorem 1.36. - Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. <a href="https://archive.org/details/functionalanalys00rudi" target="_blank">https://archive.org/details/functionalanalys00rudi</a> <a href="#fnref:51" class="footnote-back-ref">↩</a></li>
<li id="fn:52">Rudin 1991, p. 9 §1.8. - Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. <a href="https://archive.org/details/functionalanalys00rudi" target="_blank">https://archive.org/details/functionalanalys00rudi</a> <a href="#fnref:52" class="footnote-back-ref">↩</a></li>
<li id="fn:53">Rudin 1991, p. 62-68 §3.8-3.14. - Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. <a href="https://archive.org/details/functionalanalys00rudi" target="_blank">https://archive.org/details/functionalanalys00rudi</a> <a href="#fnref:53" class="footnote-back-ref">↩</a></li>
<li id="fn:54">Narici & Beckenstein 2011, pp. 177–220. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:54" class="footnote-back-ref">↩</a></li>
<li id="fn:55">Rudin 1991, p. 38. - Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. <a href="https://archive.org/details/functionalanalys00rudi" target="_blank">https://archive.org/details/functionalanalys00rudi</a> <a href="#fnref:55" class="footnote-back-ref">↩</a></li>
<li id="fn:56">Rudin 1991, p. 38. - Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. <a href="https://archive.org/details/functionalanalys00rudi" target="_blank">https://archive.org/details/functionalanalys00rudi</a> <a href="#fnref:56" class="footnote-back-ref">↩</a></li>
<li id="fn:57">Narici & Beckenstein 2011, pp. 67–113. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:57" class="footnote-back-ref">↩</a></li>
<li id="fn:58">Schaefer & Wolff 1999, p. 35. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:58" class="footnote-back-ref">↩</a></li>
<li id="fn:59">Narici & Beckenstein 2011, pp. 67–113. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:59" class="footnote-back-ref">↩</a></li>
<li id="fn:60">Narici & Beckenstein 2011, pp. 67–113. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:60" class="footnote-back-ref">↩</a></li>
<li id="fn:61">Narici & Beckenstein 2011, pp. 177–220. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:61" class="footnote-back-ref">↩</a></li>
<li id="fn:62">Narici & Beckenstein 2011, p. 119-120. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:62" class="footnote-back-ref">↩</a></li>
<li id="fn:63">Wilansky 2013, p. 43. - Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114. <a href="https://search.worldcat.org/oclc/849801114" target="_blank">https://search.worldcat.org/oclc/849801114</a> <a href="#fnref:63" class="footnote-back-ref">↩</a></li>
<li id="fn:64">Wilansky 2013, p. 42. - Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114. <a href="https://search.worldcat.org/oclc/849801114" target="_blank">https://search.worldcat.org/oclc/849801114</a> <a href="#fnref:64" class="footnote-back-ref">↩</a></li>
<li id="fn:65">Rudin 1991, p. 55. - Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. <a href="https://archive.org/details/functionalanalys00rudi" target="_blank">https://archive.org/details/functionalanalys00rudi</a> <a href="#fnref:65" class="footnote-back-ref">↩</a></li>
<li id="fn:66">Rudin 1991, p. 55. - Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. <a href="https://archive.org/details/functionalanalys00rudi" target="_blank">https://archive.org/details/functionalanalys00rudi</a> <a href="#fnref:66" class="footnote-back-ref">↩</a></li>
<li id="fn:67">Narici & Beckenstein 2011, p. 108. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:67" class="footnote-back-ref">↩</a></li>
<li id="fn:68">Narici & Beckenstein 2011, pp. 67–113. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:68" class="footnote-back-ref">↩</a></li>
<li id="fn:69">This is because every non-empty balanced set must contain the origin and because 
 
 
 
 0
 ∈
 
 Int
 
 X
 
 
 ⁡
 S
 
 
 {\displaystyle 0\in \operatorname {Int} _{X}S}
 
 if and only if 
 
 
 
 
 Int
 
 X
 
 
 ⁡
 S
 =
 {
 0
 }
 ∪
 
 Int
 
 X
 
 
 ⁡
 S
 .
 
 
 {\displaystyle \operatorname {Int} _{X}S=\{0\}\cup \operatorname {Int} _{X}S.}
 <a href="#fnref:69" class="footnote-back-ref">↩</a></li>
<li id="fn:70">Narici & Beckenstein 2011, pp. 67–113. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:70" class="footnote-back-ref">↩</a></li>
<li id="fn:71">Narici & Beckenstein 2011, p. 108. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:71" class="footnote-back-ref">↩</a></li>
<li id="fn:72">Jarchow 1981, pp. 101–104. - Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342. <a href="https://search.worldcat.org/oclc/8210342" target="_blank">https://search.worldcat.org/oclc/8210342</a> <a href="#fnref:72" class="footnote-back-ref">↩</a></li>
<li id="fn:73">Schaefer & Wolff 1999, p. 38. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:73" class="footnote-back-ref">↩</a></li>
<li id="fn:74">Conway 1990, p. 102. - Conway, John B. (1990). A Course in Functional Analysis. Graduate Texts in Mathematics. Vol. 96 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908. <a href="https://search.worldcat.org/oclc/21195908" target="_blank">https://search.worldcat.org/oclc/21195908</a> <a href="#fnref:74" class="footnote-back-ref">↩</a></li>
<li id="fn:75">Fix 
 
 
 
 0
 <
 r
 <
 1
 
 
 {\displaystyle 0<r<1}
 
 so it remains to show that 
 
 
 
 
 w
 
 0
 
 
  
 
 
 
 
 =
 
 
 
 
 def
 
 
 
 
 
 
  
 r
 x
 +
 (
 1
 −
 r
 )
 y
 
 
 {\displaystyle w_{0}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~rx+(1-r)y}
 
 belongs to 
 
 
 
 
 int
 
 X
 
 
 ⁡
 C
 .
 
 
 {\displaystyle \operatorname {int} _{X}C.}
 
 By replacing 
 
 
 
 C
 ,
 x
 ,
 y
 
 
 {\displaystyle C,x,y}
 
 with 
 
 
 
 C
 −
 
 w
 
 0
 
 
 ,
 x
 −
 
 w
 
 0
 
 
 ,
 y
 −
 
 w
 
 0
 
 
 
 
 {\displaystyle C-w_{0},x-w_{0},y-w_{0}}
 
 if necessary, we may assume without loss of generality that 
 
 
 
 r
 x
 +
 (
 1
 −
 r
 )
 y
 =
 0
 ,
 
 
 {\displaystyle rx+(1-r)y=0,}
 
 and so it remains to show that 
 
 
 
 C
 
 
 {\displaystyle C}
 
 is a neighborhood of the origin. Let 
 
 
 
 s
  
 
 
 
 
 =
 
 
 
 
 def
 
 
 
 
 
 
  
 
 
 
 r
 
 r
 −
 1
 
 
 
 
 <
 0
 
 
 {\displaystyle s~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~{\tfrac {r}{r-1}}<0}
 
 so that 
 
 
 
 y
 =
 
 
 
 r
 
 r
 −
 1
 
 
 
 
 x
 =
 s
 x
 .
 
 
 {\displaystyle y={\tfrac {r}{r-1}}x=sx.}
 
 Since scalar multiplication by 
 
 
 
 s
 ≠
 0
 
 
 {\displaystyle s\neq 0}
 
 is a linear homeomorphism 
 
 
 
 X
 →
 X
 ,
 
 
 {\displaystyle X\to X,}
 
 
 
 
 
 
 cl
 
 X
 
 
 ⁡
 
 (
 
 
 
 
 1
 s
 
 
 
 C
 
 )
 
 =
 
 
 
 1
 s
 
 
 
 
 cl
 
 X
 
 
 ⁡
 C
 .
 
 
 {\displaystyle \operatorname {cl} _{X}\left({\tfrac {1}{s}}C\right)={\tfrac {1}{s}}\operatorname {cl} _{X}C.}
 
 Since 
 
 
 
 x
 ∈
 int
 ⁡
 C
 
 
 {\displaystyle x\in \operatorname {int} C}
 
 and 
 
 
 
 y
 ∈
 cl
 ⁡
 C
 ,
 
 
 {\displaystyle y\in \operatorname {cl} C,}
 
 it follows that 
 
 
 
 x
 =
 
 
 
 1
 s
 
 
 
 y
 ∈
 cl
 ⁡
 
 (
 
 
 
 
 1
 s
 
 
 
 C
 
 )
 
 ∩
 int
 ⁡
 C
 
 
 {\displaystyle x={\tfrac {1}{s}}y\in \operatorname {cl} \left({\tfrac {1}{s}}C\right)\cap \operatorname {int} C}
 
 where because 
 
 
 
 int
 ⁡
 C
 
 
 {\displaystyle \operatorname {int} C}
 
 is open, there exists some 
 
 
 
 
 c
 
 0
 
 
 ∈
 
 (
 
 
 
 
 1
 s
 
 
 
 C
 
 )
 
 ∩
 int
 ⁡
 C
 ,
 
 
 {\displaystyle c_{0}\in \left({\tfrac {1}{s}}C\right)\cap \operatorname {int} C,}
 
 which satisfies 
 
 
 
 s
 
 c
 
 0
 
 
 ∈
 C
 .
 
 
 {\displaystyle sc_{0}\in C.}
 
 Define 
 
 
 
 h
 :
 X
 →
 X
 
 
 {\displaystyle h:X\to X}
 
 by 
 
 
 
 x
 ↦
 r
 x
 +
 (
 1
 −
 r
 )
 s
 
 c
 
 0
 
 
 =
 r
 x
 −
 r
 
 c
 
 0
 
 
 ,
 
 
 {\displaystyle x\mapsto rx+(1-r)sc_{0}=rx-rc_{0},}
 
 which is a homeomorphism because 
 
 
 
 0
 <
 r
 <
 1.
 
 
 {\displaystyle 0<r<1.}
 
 The set 
 
 
 
 h
 
 (
 
 int
 ⁡
 C
 
 )
 
 
 
 {\displaystyle h\left(\operatorname {int} C\right)}
 
 is thus an open subset of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 that moreover contains 
 
 
 
 h
 (
 
 c
 
 0
 
 
 )
 =
 r
 
 c
 
 0
 
 
 −
 r
 
 c
 
 0
 
 
 =
 0.
 
 
 {\textstyle h(c_{0})=rc_{0}-rc_{0}=0.}
 
 If 
 
 
 
 c
 ∈
 int
 ⁡
 C
 
 
 {\displaystyle c\in \operatorname {int} C}
 
 then 
 
 
 
 h
 (
 c
 )
 =
 r
 c
 +
 (
 1
 −
 r
 )
 s
 
 c
 
 0
 
 
 ∈
 C
 
 
 {\textstyle h(c)=rc+(1-r)sc_{0}\in C}
 
 since 
 
 
 
 C
 
 
 {\displaystyle C}
 
 is convex, 
 
 
 
 0
 <
 r
 <
 1
 ,
 
 
 {\displaystyle 0<r<1,}
 
 and 
 
 
 
 s
 
 c
 
 0
 
 
 ,
 c
 ∈
 C
 ,
 
 
 {\displaystyle sc_{0},c\in C,}
 
 which proves that 
 
 
 
 h
 
 (
 
 int
 ⁡
 C
 
 )
 
 ⊆
 C
 .
 
 
 {\displaystyle h\left(\operatorname {int} C\right)\subseteq C.}
 
 Thus 
 
 
 
 h
 
 (
 
 int
 ⁡
 C
 
 )
 
 
 
 {\displaystyle h\left(\operatorname {int} C\right)}
 
 is an open subset of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 that contains the origin and is contained in 
 
 
 
 C
 .
 
 
 {\displaystyle C.}
 
 Q.E.D. <a href="#fnref:75" class="footnote-back-ref">↩</a></li>
<li id="fn:76">Schaefer & Wolff 1999, p. 38. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:76" class="footnote-back-ref">↩</a></li>
<li id="fn:77">Narici & Beckenstein 2011, pp. 155–176. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:77" class="footnote-back-ref">↩</a></li>
<li id="fn:78">Since 
 
 
 
 
 cl
 
 X
 
 
 ⁡
 {
 0
 }
 
 
 {\displaystyle \operatorname {cl} _{X}\{0\}}
 
 has the trivial topology, so does each of its subsets, which makes them all compact. It is known that a subset of any uniform space is compact if and only if it is complete and totally bounded. <a href="#fnref:78" class="footnote-back-ref">↩</a></li>
<li id="fn:79">Narici & Beckenstein 2011, pp. 47–66. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:79" class="footnote-back-ref">↩</a></li>
<li id="fn:80">Narici & Beckenstein 2011, p. 156. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:80" class="footnote-back-ref">↩</a></li>
<li id="fn:81">In general topology, the closure of a compact subset of a non-Hausdorff space may fail to be compact (for example, the particular point topology on an infinite set). This result shows that this does not happen in non-Hausdorff TVSs. 
 
 
 
 S
 +
 
 cl
 
 X
 
 
 ⁡
 {
 0
 }
 
 
 {\displaystyle S+\operatorname {cl} _{X}\{0\}}
 
 is compact because it is the image of the compact set 
 
 
 
 S
 ×
 
 cl
 
 X
 
 
 ⁡
 {
 0
 }
 
 
 {\displaystyle S\times \operatorname {cl} _{X}\{0\}}
 
 under the continuous addition map 
 
 
 
 ⋅
 
 +
 
 ⋅
 
 :
 X
 ×
 X
 →
 X
 .
 
 
 {\displaystyle \cdot \,+\,\cdot \;:X\times X\to X.}
 
 Recall also that the sum of a compact set (that is, 
 
 
 
 S
 
 
 {\displaystyle S}
 
) and a closed set is closed so 
 
 
 
 S
 +
 
 cl
 
 X
 
 
 ⁡
 {
 0
 }
 
 
 {\displaystyle S+\operatorname {cl} _{X}\{0\}}
 
 is closed in 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 <a href="/wiki/Particular_point_topology" target="_blank">/wiki/Particular_point_topology</a> <a href="#fnref:81" class="footnote-back-ref">↩</a></li>
<li id="fn:82">If 
 
 
 
 s
 ∈
 S
 
 
 {\displaystyle s\in S}
 
 then 
 
 
 
 s
 +
 
 cl
 
 X
 
 
 ⁡
 {
 0
 }
 =
 
 cl
 
 X
 
 
 ⁡
 (
 s
 +
 {
 0
 }
 )
 =
 
 cl
 
 X
 
 
 ⁡
 {
 s
 }
 ⊆
 
 cl
 
 X
 
 
 ⁡
 S
 .
 
 
 {\displaystyle s+\operatorname {cl} _{X}\{0\}=\operatorname {cl} _{X}(s+\{0\})=\operatorname {cl} _{X}\{s\}\subseteq \operatorname {cl} _{X}S.}
 
 Because 
 
 
 
 S
 ⊆
 S
 +
 
 cl
 
 X
 
 
 ⁡
 {
 0
 }
 ⊆
 
 cl
 
 X
 
 
 ⁡
 S
 ,
 
 
 {\displaystyle S\subseteq S+\operatorname {cl} _{X}\{0\}\subseteq \operatorname {cl} _{X}S,}
 
 if 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is closed then equality holds. Using the fact that 
 
 
 
 
 cl
 
 X
 
 
 ⁡
 {
 0
 }
 
 
 {\displaystyle \operatorname {cl} _{X}\{0\}}
 
 is a vector space, it is readily verified that the complement in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 of any set 
 
 
 
 S
 
 
 {\displaystyle S}
 
 satisfying the equality 
 
 
 
 S
 +
 
 cl
 
 X
 
 
 ⁡
 {
 0
 }
 =
 S
 
 
 {\displaystyle S+\operatorname {cl} _{X}\{0\}=S}
 
 must also satisfy this equality (when 
 
 
 
 X
 ∖
 S
 
 
 {\displaystyle X\setminus S}
 
 is substituted for 
 
 
 
 S
 
 
 {\displaystyle S}
 
). <a href="#fnref:82" class="footnote-back-ref">↩</a></li>
<li id="fn:83">Schaefer & Wolff 1999, pp. 12–35. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:83" class="footnote-back-ref">↩</a></li>
<li id="fn:84">Schaefer & Wolff 1999, p. 25. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:84" class="footnote-back-ref">↩</a></li>
<li id="fn:85">Jarchow 1981, pp. 56–73. - Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342. <a href="https://search.worldcat.org/oclc/8210342" target="_blank">https://search.worldcat.org/oclc/8210342</a> <a href="#fnref:85" class="footnote-back-ref">↩</a></li>
<li id="fn:86">Schaefer & Wolff 1999, pp. 12–35. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:86" class="footnote-back-ref">↩</a></li>
<li id="fn:87">Narici & Beckenstein 2011, pp. 107–112. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:87" class="footnote-back-ref">↩</a></li>
<li id="fn:88">Wilansky 2013, p. 63. - Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114. <a href="https://search.worldcat.org/oclc/849801114" target="_blank">https://search.worldcat.org/oclc/849801114</a> <a href="#fnref:88" class="footnote-back-ref">↩</a></li>
<li id="fn:89">Schaefer & Wolff 1999, pp. 12–35. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:89" class="footnote-back-ref">↩</a></li>
<li id="fn:90">Narici & Beckenstein 2011, pp. 47–66. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:90" class="footnote-back-ref">↩</a></li>
<li id="fn:91">Narici & Beckenstein 2011, pp. 47–66. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:91" class="footnote-back-ref">↩</a></li>
<li id="fn:92">Schaefer & Wolff 1999, p. 25. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:92" class="footnote-back-ref">↩</a></li>
<li id="fn:93">Jarchow 1981, pp. 56–73. - Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342. <a href="https://search.worldcat.org/oclc/8210342" target="_blank">https://search.worldcat.org/oclc/8210342</a> <a href="#fnref:93" class="footnote-back-ref">↩</a></li>
<li id="fn:94">Schaefer & Wolff 1999, pp. 12–35. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:94" class="footnote-back-ref">↩</a></li>
<li id="fn:95">Narici & Beckenstein 2011, pp. 47–66. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:95" class="footnote-back-ref">↩</a></li>
<li id="fn:96">Narici & Beckenstein 2011, pp. 47–66. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:96" class="footnote-back-ref">↩</a></li>
<li id="fn:97">Narici & Beckenstein 2011, pp. 47–66. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:97" class="footnote-back-ref">↩</a></li>
<li id="fn:98">Schaefer & Wolff 1999, pp. 12–35. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:98" class="footnote-back-ref">↩</a></li>
<li id="fn:99">Narici & Beckenstein 2011, pp. 19–45. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:99" class="footnote-back-ref">↩</a></li>
<li id="fn:100">Narici & Beckenstein 2011, pp. 67–113. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:100" class="footnote-back-ref">↩</a></li>
<li id="fn:101">Narici & Beckenstein 2011, pp. 19–45. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:101" class="footnote-back-ref">↩</a></li>
<li id="fn:102">Narici & Beckenstein 2011, pp. 67–113. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:102" class="footnote-back-ref">↩</a></li>
<li id="fn:103">In 
 
 
 
 
 
 R
 
 
 2
 
 
 ,
 
 
 {\displaystyle \mathbb {R} ^{2},}
 
 the sum of the 
 
 
 
 y
 
 
 {\displaystyle y}
 
-axis and the graph of 
 
 
 
 y
 =
 
 
 1
 x
 
 
 ,
 
 
 {\displaystyle y={\frac {1}{x}},}
 
 which is the complement of the 
 
 
 
 y
 
 
 {\displaystyle y}
 
-axis, is open in 
 
 
 
 
 
 R
 
 
 2
 
 
 .
 
 
 {\displaystyle \mathbb {R} ^{2}.}
 
 In 
 
 
 
 
 R
 
 ,
 
 
 {\displaystyle \mathbb {R} ,}
 
 the Minkowski sum 
 
 
 
 
 Z
 
 +
 
 
 2
 
 
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} +{\sqrt {2}}\mathbb {Z} }
 
 is a countable dense subset of 
 
 
 
 
 R
 
 
 
 {\displaystyle \mathbb {R} }
 
 so not closed in 
 
 
 
 
 R
 
 .
 
 
 {\displaystyle \mathbb {R} .}
 <a href="/wiki/Minkowski_sum" target="_blank">/wiki/Minkowski_sum</a> <a href="#fnref:103" class="footnote-back-ref">↩</a></li>
<li id="fn:104">Wilansky 2013, pp. 43–44. - Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114. <a href="https://search.worldcat.org/oclc/849801114" target="_blank">https://search.worldcat.org/oclc/849801114</a> <a href="#fnref:104" class="footnote-back-ref">↩</a></li>
<li id="fn:105">Wilansky 2013, pp. 43–44. - Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114. <a href="https://search.worldcat.org/oclc/849801114" target="_blank">https://search.worldcat.org/oclc/849801114</a> <a href="#fnref:105" class="footnote-back-ref">↩</a></li>
<li id="fn:106">Narici & Beckenstein 2011, pp. 67–113. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:106" class="footnote-back-ref">↩</a></li>
<li id="fn:107">Wilansky 2013, pp. 43–44. - Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114. <a href="https://search.worldcat.org/oclc/849801114" target="_blank">https://search.worldcat.org/oclc/849801114</a> <a href="#fnref:107" class="footnote-back-ref">↩</a></li>
<li id="fn:108">Narici & Beckenstein 2011, pp. 80. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:108" class="footnote-back-ref">↩</a></li>
<li id="fn:109">Narici & Beckenstein 2011, pp. 108–109. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:109" class="footnote-back-ref">↩</a></li>
<li id="fn:110">Jarchow 1981, pp. 30–32. - Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342. <a href="https://search.worldcat.org/oclc/8210342" target="_blank">https://search.worldcat.org/oclc/8210342</a> <a href="#fnref:110" class="footnote-back-ref">↩</a></li>
<li id="fn:111">Narici & Beckenstein 2011, pp. 155–176. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:111" class="footnote-back-ref">↩</a></li>
<li id="fn:112">Narici & Beckenstein 2011, pp. 67–113. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:112" class="footnote-back-ref">↩</a></li>
<li id="fn:113">Narici & Beckenstein 2011, pp. 67–113. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:113" class="footnote-back-ref">↩</a></li>
<li id="fn:114">Narici & Beckenstein 2011, p. 109. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:114" class="footnote-back-ref">↩</a></li>
<li id="fn:115">Narici & Beckenstein 2011, p. 109. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:115" class="footnote-back-ref">↩</a></li>
<li id="fn:116">Narici & Beckenstein 2011, p. 109. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:116" class="footnote-back-ref">↩</a></li>
<li id="fn:117">Narici & Beckenstein 2011, pp. 67–113. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:117" class="footnote-back-ref">↩</a></li>
<li id="fn:118">Narici & Beckenstein 2011, pp. 67–113. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:118" class="footnote-back-ref">↩</a></li>
<li id="fn:119">Narici & Beckenstein 2011, pp. 371–423. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:119" class="footnote-back-ref">↩</a></li>
<li id="fn:120">Narici & Beckenstein 2011, pp. 371–423. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:120" class="footnote-back-ref">↩</a></li>
<li id="fn:121">Narici & Beckenstein 2011, pp. 371–423. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:121" class="footnote-back-ref">↩</a></li>
<li id="fn:122">Narici & Beckenstein 2011, pp. 371–423. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:122" class="footnote-back-ref">↩</a></li>
<li id="fn:123">Narici & Beckenstein 2011, pp. 371–423. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:123" class="footnote-back-ref">↩</a></li>
<li id="fn:124">Rudin 1991, p. 38. - Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. <a href="https://archive.org/details/functionalanalys00rudi" target="_blank">https://archive.org/details/functionalanalys00rudi</a> <a href="#fnref:124" class="footnote-back-ref">↩</a></li>
<li id="fn:125">Rudin 1991, p. 6. - Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. <a href="https://archive.org/details/functionalanalys00rudi" target="_blank">https://archive.org/details/functionalanalys00rudi</a> <a href="#fnref:125" class="footnote-back-ref">↩</a></li>
<li id="fn:126">Narici & Beckenstein 2011, pp. 67–113. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:126" class="footnote-back-ref">↩</a></li>
<li id="fn:127">Swartz 1992, p. 35. - Swartz, Charles (1992). An introduction to Functional Analysis. New York: M. Dekker. ISBN 978-0-8247-8643-4. OCLC 24909067. <a href="https://search.worldcat.org/oclc/24909067" target="_blank">https://search.worldcat.org/oclc/24909067</a> <a href="#fnref:127" class="footnote-back-ref">↩</a></li>
<li id="fn:128">z
 +
 {
 x
 ∈
 X
 :
 P
 (
 x
 )
 }
 =
 {
 z
 +
 x
 :
 x
 ∈
 X
 ,
 P
 (
 x
 )
 }
 =
 {
 z
 +
 x
 :
 x
 ∈
 X
 ,
 P
 (
 (
 z
 +
 x
 )
 −
 z
 )
 }
 
 
 {\displaystyle z+\{x\in X:P(x)\}=\{z+x:x\in X,P(x)\}=\{z+x:x\in X,P((z+x)-z)\}}
 
 and so using 
 
 
 
 y
 =
 z
 +
 x
 
 
 {\displaystyle y=z+x}
 
 and the fact that 
 
 
 
 z
 +
 X
 =
 X
 ,
 
 
 {\displaystyle z+X=X,}
 
 this is equal to 
 
 
 
 {
 y
 :
 y
 −
 z
 ∈
 X
 ,
 P
 (
 y
 −
 z
 )
 }
 =
 {
 y
 :
 y
 ∈
 X
 ,
 P
 (
 y
 −
 z
 )
 }
 =
 {
 y
 ∈
 X
 :
 P
 (
 y
 −
 z
 )
 }
 .
 
 
 {\displaystyle \{y:y-z\in X,P(y-z)\}=\{y:y\in X,P(y-z)\}=\{y\in X:P(y-z)\}.}
 
 Q.E.D. 
 
 
 
 ◼
 
 
 {\displaystyle \blacksquare }
 <a href="/wiki/Q.E.D." target="_blank">/wiki/Q.E.D.</a> <a href="#fnref:128" class="footnote-back-ref">↩</a></li>
<li id="fn:129">Narici & Beckenstein 2011, pp. 67–113. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:129" class="footnote-back-ref">↩</a></li>
<li id="fn:130">Narici & Beckenstein 2011, pp. 67–113. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:130" class="footnote-back-ref">↩</a></li>
<li id="fn:131">Narici & Beckenstein 2011, pp. 67–113. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:131" class="footnote-back-ref">↩</a></li>
</ol>

Topological vector space open-in-new

Topological vector space