Auxiliary normed space

<h2 id="induced-by-a-bounded-disk--banach-disks">Induced by a bounded disk – Banach disks</h2>
Throughout this article, 
 
 
 
 X
 
 
 {\displaystyle X}
 
 will be a real or complex vector space (not necessarily a TVS, yet) and 
 
 
 
 D
 
 
 {\displaystyle D}
 
 will be a <a href="/facts/Absolutely_convex_set/rJ5HLWkg">disk</a> in 
 
 
 
 X
 .
 
 
 {\displaystyle X.}

<h3>Seminormed space induced by a disk</h3>
Let 
 
 
 
 X
 
 
 {\displaystyle X}
 
 will be a real or complex vector space. For any subset 
 
 
 
 D
 
 
 {\displaystyle D}
 
 of 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 the <a href="/facts/Minkowski_functional/1qoFAjRs">Minkowski functional</a> of 
 
 
 
 D
 
 
 {\displaystyle D}
 
 defined by:

<ul><li>If 
 
 
 
 D
 =
 ∅
 
 
 {\displaystyle D=\varnothing }
 
 then define 
 
 
 
 
 p
 
 ∅
 
 
 (
 x
 )
 :
 {
 0
 }
 →
 [
 0
 ,
 ∞
 )
 
 
 {\displaystyle p_{\varnothing }(x):\{0\}\to [0,\infty )}
 
 to be the trivial map 
 
 
 
 
 p
 
 ∅
 
 
 =
 0
 
 
 {\displaystyle p_{\varnothing }=0}
 
<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> and it will be assumed that 
 
 
 
 span
 ⁡
 ∅
 =
 {
 0
 }
 .
 
 
 {\displaystyle \operatorname {span} \varnothing =\{0\}.}
 
<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a></li>
<li>If 
 
 
 
 D
 ≠
 ∅
 
 
 {\displaystyle D\neq \varnothing }
 
 and if 
 
 
 
 D
 
 
 {\displaystyle D}
 
 is <a href="/facts/Absorbing_set/GCtjsUuX">absorbing</a> in 
 
 
 
 span
 ⁡
 D
 
 
 {\displaystyle \operatorname {span} D}
 
 then denote the <a href="/facts/Minkowski_functional/1qoFAjRs">Minkowski functional</a> of 
 
 
 
 D
 
 
 {\displaystyle D}
 
 in 
 
 
 
 span
 ⁡
 D
 
 
 {\displaystyle \operatorname {span} D}
 
 by 
 
 
 
 
 p
 
 D
 
 
 :
 span
 ⁡
 D
 →
 [
 0
 ,
 ∞
 )
 
 
 {\displaystyle p_{D}:\operatorname {span} D\to [0,\infty )}
 
 where for all 
 
 
 
 x
 ∈
 span
 ⁡
 D
 ,
 
 
 {\displaystyle x\in \operatorname {span} D,}
 
 this is defined by 
 
 
 
 
 p
 
 D
 
 
 (
 x
 )
 :=
 
 inf

{
 r
 :
 x
 ∈
 r
 D
 ,
 r
 >
 0
 }
 .
 
 
 {\displaystyle p_{D}(x):=\inf _{}\{r:x\in rD,r>0\}.}
 
</li></ul>
Let 
 
 
 
 X
 
 
 {\displaystyle X}
 
 will be a real or complex vector space. For any subset 
 
 
 
 D
 
 
 {\displaystyle D}
 
 of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 such that the Minkowski functional 
 
 
 
 
 p
 
 D
 
 
 
 
 {\displaystyle p_{D}}
 
 is a <a href="/facts/Seminorm/vMuescKp">seminorm</a> on 
 
 
 
 span
 ⁡
 D
 ,
 
 
 {\displaystyle \operatorname {span} D,}
 
 let 
 
 
 
 
 X
 
 D
 
 
 
 
 {\displaystyle X_{D}}
 
 denote

X
          
            D
          
        
        :=
        
          (
          
            span
            ⁡
            D
            ,
            
              p
              
                D
              
            
          
          )
        
      
    
    {\displaystyle X_{D}:=\left(\operatorname {span} D,p_{D}\right)}

which is called the <a href="/facts/Seminormed_space/vMuescKp">seminormed space</a> induced by 
 
 
 
 D
 ,
 
 
 {\displaystyle D,}
 
 where if 
 
 
 
 
 p
 
 D
 
 
 
 
 {\displaystyle p_{D}}
 
 is a <a href="/facts/Norm_(mathematics)/xIbR4uE1">norm</a> then it is called the <a href="/facts/Normed_space/rmDljfyE">normed space</a> induced by 
 
 
 
 D
 .
 
 
 {\displaystyle D.}
 

Assumption (Topology): 
 
 
 
 
 X
 
 D
 
 
 =
 span
 ⁡
 D
 
 
 {\displaystyle X_{D}=\operatorname {span} D}
 
 is endowed with the seminorm topology induced by 
 
 
 
 
 p
 
 D
 
 
 ,
 
 
 {\displaystyle p_{D},}
 
 which will be denoted by 
 
 
 
 
 τ
 
 D
 
 
 
 
 {\displaystyle \tau _{D}}
 
 or 
 
 
 
 
 τ
 
 
 p
 
 D
 
 
 
 
 
 
 {\displaystyle \tau _{p_{D}}}

Importantly, this topology stems entirely from the set 
 
 
 
 D
 ,
 
 
 {\displaystyle D,}
 
 the algebraic structure of 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 and the usual topology on 
 
 
 
 
 R
 
 
 
 {\displaystyle \mathbb {R} }
 
 (since 
 
 
 
 
 p
 
 D
 
 
 
 
 {\displaystyle p_{D}}
 
 is defined using only the set 
 
 
 
 D
 
 
 {\displaystyle D}
 
 and scalar multiplication). This justifies the study of Banach disks and is part of the reason why they play an important role in the theory of <a href="/facts/Nuclear_operator/kxnx5bIJ">nuclear operators</a> and <a href="/facts/Nuclear_space/4FLPWdfp">nuclear spaces</a>.
The inclusion map 
 
 
 
 
 In
 
 D
 
 
 :
 
 X
 
 D
 
 
 →
 X
 
 
 {\displaystyle \operatorname {In} _{D}:X_{D}\to X}
 
 is called the canonical map.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>
Suppose that 
 
 
 
 D
 
 
 {\displaystyle D}
 
 is a disk. 
Then 
 
 
 
 span
 ⁡
 D
 =
 
 ⋃
 
 n
 =
 1
 
 
 ∞
 
 
 n
 D
 
 
 {\textstyle \operatorname {span} D=\bigcup _{n=1}^{\infty }nD}
 
 so that 
 
 
 
 D
 
 
 {\displaystyle D}
 
 is <a href="/facts/Absorbing_set/GCtjsUuX">absorbing</a> in 
 
 
 
 span
 ⁡
 D
 ,
 
 
 {\displaystyle \operatorname {span} D,}
 
 the <a href="/facts/Linear_span/vPjclEtb">linear span</a> of 
 
 
 
 D
 .
 
 
 {\displaystyle D.}
 
 
The set 
 
 
 
 {
 r
 D
 :
 r
 >
 0
 }
 
 
 {\displaystyle \{rD:r>0\}}
 
 of all positive scalar multiples of 
 
 
 
 D
 
 
 {\displaystyle D}
 
 forms a basis of neighborhoods at the origin for a <a href="/facts/Locally_convex_topological_vector_space/AGrPrayC">locally convex topological vector space</a> topology 
 
 
 
 
 τ
 
 D
 
 
 
 
 {\displaystyle \tau _{D}}
 
 on 
 
 
 
 span
 ⁡
 D
 .
 
 
 {\displaystyle \operatorname {span} D.}
 
 
The <a href="/facts/Minkowski_functional/1qoFAjRs">Minkowski functional</a> of the disk 
 
 
 
 D
 
 
 {\displaystyle D}
 
 in 
 
 
 
 span
 ⁡
 D
 
 
 {\displaystyle \operatorname {span} D}
 
 guarantees that 
 
 
 
 
 p
 
 D
 
 
 
 
 {\displaystyle p_{D}}
 
 is well-defined and forms a <a href="/facts/Seminorm/vMuescKp">seminorm</a> on 
 
 
 
 span
 ⁡
 D
 .
 
 
 {\displaystyle \operatorname {span} D.}
 
<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> 
The locally convex topology induced by this seminorm is the topology 
 
 
 
 
 τ
 
 D
 
 
 
 
 {\displaystyle \tau _{D}}
 
 that was defined before.

<h3>Banach disk definition</h3>
A <a href="/facts/Bounded_set_(topological_vector_space)/bTsWeU9g">bounded</a> <a href="/facts/Absolutely_convex_set/rJ5HLWkg">disk</a> 
 
 
 
 D
 
 
 {\displaystyle D}
 
 in a <a href="/facts/Topological_vector_space/XRex1ChN">topological vector space</a> 
 
 
 
 X
 
 
 {\displaystyle X}
 
 such that 
 
 
 
 
 (
 
 
 X
 
 D
 
 
 ,
 
 p
 
 D
 
 
 
 )
 
 
 
 {\displaystyle \left(X_{D},p_{D}\right)}
 
 is a <a href="/facts/Banach_space/sDEIk7EC">Banach space</a> is called a Banach disk, infracomplete, or a bounded completant in 
 
 
 
 X
 .
 
 
 {\displaystyle X.}

If its shown that 
 
 
 
 
 (
 
 span
 ⁡
 D
 ,
 
 p
 
 D
 
 
 
 )
 
 
 
 {\displaystyle \left(\operatorname {span} D,p_{D}\right)}
 
 is a Banach space then 
 
 
 
 D
 
 
 {\displaystyle D}
 
 will be a Banach disk in any TVS that contains 
 
 
 
 D
 
 
 {\displaystyle D}
 
 as a bounded subset.
This is because the Minkowski functional 
 
 
 
 
 p
 
 D
 
 
 
 
 {\displaystyle p_{D}}
 
 is defined in purely algebraic terms. 
Consequently, the question of whether or not 
 
 
 
 
 (
 
 
 X
 
 D
 
 
 ,
 
 p
 
 D
 
 
 
 )
 
 
 
 {\displaystyle \left(X_{D},p_{D}\right)}
 
 forms a Banach space is dependent only on the disk 
 
 
 
 D
 
 
 {\displaystyle D}
 
 and the Minkowski functional 
 
 
 
 
 p
 
 D
 
 
 ,
 
 
 {\displaystyle p_{D},}
 
 and not on any particular TVS topology that 
 
 
 
 X
 
 
 {\displaystyle X}
 
 may carry. 
Thus the requirement that a Banach disk in a TVS 
 
 
 
 X
 
 
 {\displaystyle X}
 
 be a bounded subset of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is the only property that ties a Banach disk's topology to the topology of its containing TVS 
 
 
 
 X
 .
 
 
 {\displaystyle X.}

<h3>Properties of disk induced seminormed spaces</h3>
Bounded disks
The following result explains why Banach disks are required to be bounded.

Theorem<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a><a class="footnote-ref" id="fnref:7" href="#fn:7">7</a><a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>—If 
 
 
 
 D
 
 
 {\displaystyle D}
 
 is a disk in a <a href="/facts/Topological_vector_space/XRex1ChN">topological vector space</a> (TVS) 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 then 
 
 
 
 D
 
 
 {\displaystyle D}
 
 is <a href="/facts/Bounded_set_(topological_vector_space)/bTsWeU9g">bounded</a> in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 if and only if the inclusion map 
 
 
 
 
 In
 
 D
 
 
 :
 
 X
 
 D
 
 
 →
 X
 
 
 {\displaystyle \operatorname {In} _{D}:X_{D}\to X}
 
 is continuous.

Proof
If the disk 
 
 
 
 D
 
 
 {\displaystyle D}
 
 is bounded in the TVS 
 
 
 
 X
 
 
 {\displaystyle X}
 
 then for all neighborhoods 
 
 
 
 U
 
 
 {\displaystyle U}
 
 of the origin in 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 there exists some 
 
 
 
 r
 >
 0
 
 
 {\displaystyle r>0}
 
 such that 
 
 
 
 r
 D
 ⊆
 U
 ∩
 
 X
 
 D
 
 
 .
 
 
 {\displaystyle rD\subseteq U\cap X_{D}.}
 
 
It follows that in this case the topology of 
 
 
 
 
 (
 
 
 X
 
 D
 
 
 ,
 
 p
 
 D
 
 
 
 )
 
 
 
 {\displaystyle \left(X_{D},p_{D}\right)}
 
 is finer than the subspace topology that 
 
 
 
 
 X
 
 D
 
 
 
 
 {\displaystyle X_{D}}
 
 inherits from 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 which implies that the inclusion map 
 
 
 
 
 In
 
 D
 
 
 :
 
 X
 
 D
 
 
 →
 X
 
 
 {\displaystyle \operatorname {In} _{D}:X_{D}\to X}
 
 is continuous. 
Conversely, if 
 
 
 
 X
 
 
 {\displaystyle X}
 
 has a TVS topology such that 
 
 
 
 
 In
 
 D
 
 
 :
 
 X
 
 D
 
 
 →
 X
 
 
 {\displaystyle \operatorname {In} _{D}:X_{D}\to X}
 
 is continuous, then for every neighborhood 
 
 
 
 U
 
 
 {\displaystyle U}
 
 of the origin in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 there exists some 
 
 
 
 r
 >
 0
 
 
 {\displaystyle r>0}
 
 such that 
 
 
 
 r
 D
 ⊆
 U
 ∩
 
 X
 
 D
 
 
 ,
 
 
 {\displaystyle rD\subseteq U\cap X_{D},}
 
 which shows that 
 
 
 
 D
 
 
 {\displaystyle D}
 
 is bounded in 
 
 
 
 X
 .
 
 
 {\displaystyle X.}

Hausdorffness
The space 
 
 
 
 
 (
 
 
 X
 
 D
 
 
 ,
 
 p
 
 D
 
 
 
 )
 
 
 
 {\displaystyle \left(X_{D},p_{D}\right)}
 
 is <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff</a> if and only if 
 
 
 
 
 p
 
 D
 
 
 
 
 {\displaystyle p_{D}}
 
 is a norm, which happens if and only if 
 
 
 
 D
 
 
 {\displaystyle D}
 
 does not contain any non-trivial vector subspace.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> 
In particular, if there exists a Hausdorff TVS topology on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 such that 
 
 
 
 D
 
 
 {\displaystyle D}
 
 is bounded in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 then 
 
 
 
 
 p
 
 D
 
 
 
 
 {\displaystyle p_{D}}
 
 is a norm. 
An example where 
 
 
 
 
 X
 
 D
 
 
 
 
 {\displaystyle X_{D}}
 
 is not Hausdorff is obtained by letting 
 
 
 
 X
 =
 
 
 R
 
 
 2
 
 
 
 
 {\displaystyle X=\mathbb {R} ^{2}}
 
 and letting 
 
 
 
 D
 
 
 {\displaystyle D}
 
 be the 
 
 
 
 x
 
 
 {\displaystyle x}
 
-axis.
Convergence of nets
Suppose that 
 
 
 
 D
 
 
 {\displaystyle D}
 
 is a disk in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 such that 
 
 
 
 
 X
 
 D
 
 
 
 
 {\displaystyle X_{D}}
 
 is Hausdorff and let 
 
 
 
 
 x
 
 ∙
 
 
 =
 
 
 (
 
 x
 
 i
 
 
 )
 
 
 i
 ∈
 I
 
 
 
 
 {\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}
 
 be a net in 
 
 
 
 
 X
 
 D
 
 
 .
 
 
 {\displaystyle X_{D}.}
 
 
Then 
 
 
 
 
 x
 
 ∙
 
 
 →
 0
 
 
 {\displaystyle x_{\bullet }\to 0}
 
 in 
 
 
 
 
 X
 
 D
 
 
 
 
 {\displaystyle X_{D}}
 
 if and only if there exists a net 
 
 
 
 
 r
 
 ∙
 
 
 =
 
 
 (
 
 r
 
 i
 
 
 )
 
 
 i
 ∈
 I
 
 
 
 
 {\displaystyle r_{\bullet }=\left(r_{i}\right)_{i\in I}}
 
 of real numbers such that 
 
 
 
 
 r
 
 ∙
 
 
 →
 0
 
 
 {\displaystyle r_{\bullet }\to 0}
 
 and 
 
 
 
 
 x
 
 i
 
 
 ∈
 
 r
 
 i
 
 
 D
 
 
 {\displaystyle x_{i}\in r_{i}D}
 
 for all 
 
 
 
 i
 
 
 {\displaystyle i}
 
; 
moreover, in this case it will be assumed without loss of generality that 
 
 
 
 
 r
 
 i
 
 
 ≥
 0
 
 
 {\displaystyle r_{i}\geq 0}
 
 for all 
 
 
 
 i
 .
 
 
 {\displaystyle i.}

Relationship between disk-induced spaces
If 
 
 
 
 C
 ⊆
 D
 ⊆
 X
 
 
 {\displaystyle C\subseteq D\subseteq X}
 
 then 
 
 
 
 span
 ⁡
 C
 ⊆
 span
 ⁡
 D
 
 
 {\displaystyle \operatorname {span} C\subseteq \operatorname {span} D}
 
 and 
 
 
 
 
 p
 
 D
 
 
 ≤
 
 p
 
 C
 
 
 
 
 {\displaystyle p_{D}\leq p_{C}}
 
 on 
 
 
 
 span
 ⁡
 C
 ,
 
 
 {\displaystyle \operatorname {span} C,}
 
 so define the following continuous<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> linear map:
If 
 
 
 
 C
 
 
 {\displaystyle C}
 
 and 
 
 
 
 D
 
 
 {\displaystyle D}
 
 are disks in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 with 
 
 
 
 C
 ⊆
 D
 
 
 {\displaystyle C\subseteq D}
 
 then call the inclusion map 
 
 
 
 
 In
 
 C
 
 
 D
 
 
 :
 
 X
 
 C
 
 
 →
 
 X
 
 D
 
 
 
 
 {\displaystyle \operatorname {In} _{C}^{D}:X_{C}\to X_{D}}
 
 the canonical inclusion of 
 
 
 
 
 X
 
 C
 
 
 
 
 {\displaystyle X_{C}}
 
 into 
 
 
 
 
 X
 
 D
 
 
 .
 
 
 {\displaystyle X_{D}.}

In particular, the subspace topology that 
 
 
 
 span
 ⁡
 C
 
 
 {\displaystyle \operatorname {span} C}
 
 inherits from 
 
 
 
 
 (
 
 
 X
 
 D
 
 
 ,
 
 p
 
 D
 
 
 
 )
 
 
 
 {\displaystyle \left(X_{D},p_{D}\right)}
 
 is weaker than 
 
 
 
 
 (
 
 
 X
 
 C
 
 
 ,
 
 p
 
 C
 
 
 
 )
 
 
 
 {\displaystyle \left(X_{C},p_{C}\right)}
 
's seminorm topology.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>
The disk as the closed unit ball
The disk 
 
 
 
 D
 
 
 {\displaystyle D}
 
 is a closed subset of 
 
 
 
 
 (
 
 
 X
 
 D
 
 
 ,
 
 p
 
 D
 
 
 
 )
 
 
 
 {\displaystyle \left(X_{D},p_{D}\right)}
 
 if and only if 
 
 
 
 D
 
 
 {\displaystyle D}
 
 is the closed unit ball of the seminorm 
 
 
 
 
 p
 
 D
 
 
 
 
 {\displaystyle p_{D}}
 
; that is,

D
 =
 
 {
 
 x
 ∈
 span
 ⁡
 D
 :
 
 p
 
 D
 
 
 (
 x
 )
 ≤
 1
 
 }
 
 .
 
 
 {\displaystyle D=\left\{x\in \operatorname {span} D:p_{D}(x)\leq 1\right\}.}
 
 
If 
 
 
 
 D
 
 
 {\displaystyle D}
 
 is a disk in a vector space 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and if there exists a TVS topology 
 
 
 
 τ
 
 
 {\displaystyle \tau }
 
 on 
 
 
 
 span
 ⁡
 D
 
 
 {\displaystyle \operatorname {span} D}
 
 such that 
 
 
 
 D
 
 
 {\displaystyle D}
 
 is a closed and bounded subset of 
 
 
 
 
 (
 
 span
 ⁡
 D
 ,
 τ
 
 )
 
 ,
 
 
 {\displaystyle \left(\operatorname {span} D,\tau \right),}
 
 then 
 
 
 
 D
 
 
 {\displaystyle D}
 
 is the closed unit ball of 
 
 
 
 
 (
 
 
 X
 
 D
 
 
 ,
 
 p
 
 D
 
 
 
 )
 
 
 
 {\displaystyle \left(X_{D},p_{D}\right)}
 
 (that is, 
 
 
 
 D
 =
 
 {
 
 x
 ∈
 span
 ⁡
 D
 :
 
 p
 
 D
 
 
 (
 x
 )
 ≤
 1
 
 }
 
 
 
 {\displaystyle D=\left\{x\in \operatorname {span} D:p_{D}(x)\leq 1\right\}}
 
 ) (see footnote for proof).<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a>

<h3>Sufficient conditions for a Banach disk</h3>
The following theorem may be used to establish that 
 
 
 
 
 (
 
 
 X
 
 D
 
 
 ,
 
 p
 
 D
 
 
 
 )
 
 
 
 {\displaystyle \left(X_{D},p_{D}\right)}
 
 is a Banach space. 
Once this is established, 
 
 
 
 D
 
 
 {\displaystyle D}
 
 will be a Banach disk in any TVS in which 
 
 
 
 D
 
 
 {\displaystyle D}
 
 is bounded.

Theorem<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a>—Let 
 
 
 
 D
 
 
 {\displaystyle D}
 
 be a disk in a vector space 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 
 
If there exists a Hausdorff TVS topology 
 
 
 
 τ
 
 
 {\displaystyle \tau }
 
 on 
 
 
 
 span
 ⁡
 D
 
 
 {\displaystyle \operatorname {span} D}
 
 such that 
 
 
 
 D
 
 
 {\displaystyle D}
 
 is a bounded <a href="/facts/Sequentially_complete/8avaV5Uu">sequentially complete</a> subset of 
 
 
 
 (
 span
 ⁡
 D
 ,
 τ
 )
 ,
 
 
 {\displaystyle (\operatorname {span} D,\tau ),}
 
 then 
 
 
 
 
 (
 
 
 X
 
 D
 
 
 ,
 
 p
 
 D
 
 
 
 )
 
 
 
 {\displaystyle \left(X_{D},p_{D}\right)}
 
 is a Banach space.

Proof
Assume without loss of generality that 
 
 
 
 X
 =
 span
 ⁡
 D
 
 
 {\displaystyle X=\operatorname {span} D}
 
 and let 
 
 
 
 p
 :=
 
 p
 
 D
 
 
 
 
 {\displaystyle p:=p_{D}}
 
 be the <a href="/facts/Minkowski_functional/1qoFAjRs">Minkowski functional</a> of 
 
 
 
 D
 .
 
 
 {\displaystyle D.}
 
 
Since 
 
 
 
 D
 
 
 {\displaystyle D}
 
 is a bounded subset of a Hausdorff TVS, 
 
 
 
 D
 
 
 {\displaystyle D}
 
 do not contain any non-trivial vector subspace, which implies that 
 
 
 
 p
 
 
 {\displaystyle p}
 
 is a norm. 
Let 
 
 
 
 
 τ
 
 D
 
 
 
 
 {\displaystyle \tau _{D}}
 
 denote the norm topology on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 induced by 
 
 
 
 p
 
 
 {\displaystyle p}
 
 where since 
 
 
 
 D
 
 
 {\displaystyle D}
 
 is a bounded subset of 
 
 
 
 (
 X
 ,
 τ
 )
 ,
 
 
 {\displaystyle (X,\tau ),}
 
 
 
 
 
 
 τ
 
 D
 
 
 
 
 {\displaystyle \tau _{D}}
 
 is finer than 
 
 
 
 τ
 .
 
 
 {\displaystyle \tau .}

Because 
 
 
 
 D
 
 
 {\displaystyle D}
 
 is convex and balanced, for any 
 
 
 
 0
 <
 m
 <
 n
 
 
 {\displaystyle 0<m<n}

2
 
 −
 (
 n
 +
 1
 )
 
 
 D
 +
 ⋯
 +
 
 2
 
 −
 (
 m
 +
 2
 )
 
 
 D
 =
 
 2
 
 −
 (
 m
 +
 1
 )
 
 
 
 (
 
 1
 −
 
 2
 
 m
 −
 n
 
 
 
 )
 
 D
 ⊆
 
 2
 
 −
 (
 m
 +
 2
 )
 
 
 D
 .
 
 
 {\displaystyle 2^{-(n+1)}D+\cdots +2^{-(m+2)}D=2^{-(m+1)}\left(1-2^{m-n}\right)D\subseteq 2^{-(m+2)}D.}

Let 
 
 
 
 
 x
 
 ∙
 
 
 =
 
 
 (
 
 x
 
 i
 
 
 )
 
 
 i
 =
 1
 
 
 ∞
 
 
 
 
 {\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }}
 
 be a Cauchy sequence in 
 
 
 
 
 (
 
 
 X
 
 D
 
 
 ,
 p
 
 )
 
 .
 
 
 {\displaystyle \left(X_{D},p\right).}
 
 
By replacing 
 
 
 
 
 x
 
 ∙
 
 
 
 
 {\displaystyle x_{\bullet }}
 
 with a subsequence, we may assume without loss of generality† that for all 
 
 
 
 i
 ,
 
 
 {\displaystyle i,}

x
          
            i
            +
            1
          
        
        −
        
          x
          
            i
          
        
        ∈
        
          
            1
            
              2
              
                i
                +
                2
              
            
          
        
        D
        .
      
    
    {\displaystyle x_{i+1}-x_{i}\in {\frac {1}{2^{i+2}}}D.}

This implies that for any 
 
 
 
 0
 <
 m
 <
 n
 ,
 
 
 {\displaystyle 0<m<n,}

x
          
            n
          
        
        −
        
          x
          
            m
          
        
        =
        
          (
          
            
              x
              
                n
              
            
            −
            
              x
              
                n
                −
                1
              
            
          
          )
        
        +
        
          (
          
            
              x
              
                m
                +
                1
              
            
            −
            
              x
              
                m
              
            
          
          )
        
        ∈
        
          2
          
            −
            (
            n
            +
            1
            )
          
        
        D
        +
        ⋯
        +
        
          2
          
            −
            (
            m
            +
            2
            )
          
        
        D
        ⊆
        
          2
          
            −
            (
            m
            +
            2
            )
          
        
        D
      
    
    {\displaystyle x_{n}-x_{m}=\left(x_{n}-x_{n-1}\right)+\left(x_{m+1}-x_{m}\right)\in 2^{-(n+1)}D+\cdots +2^{-(m+2)}D\subseteq 2^{-(m+2)}D}

so that in particular, by taking 
  
    
      
        m
        =
        1
      
    
    {\displaystyle m=1}
  
 it follows that 
  
    
      
        
          x
          
            ∙
          
        
      
    
    {\displaystyle x_{\bullet }}
  
 is contained in 
  
    
      
        
          x
          
            1
          
        
        +
        
          2
          
            −
            3
          
        
        D
        .
      
    
    {\displaystyle x_{1}+2^{-3}D.}
  
 
Since 
  
    
      
        
          τ
          
            D
          
        
      
    
    {\displaystyle \tau _{D}}
  
 is finer than 
  
    
      
        τ
        ,
      
    
    {\displaystyle \tau ,}
  
 
  
    
      
        
          x
          
            ∙
          
        
      
    
    {\displaystyle x_{\bullet }}
  
 is a Cauchy sequence in 
  
    
      
        (
        X
        ,
        τ
        )
        .
      
    
    {\displaystyle (X,\tau ).}
  
 
For all 
  
    
      
        m
        >
        0
        ,
      
    
    {\displaystyle m>0,}
  
 
  
    
      
        
          2
          
            −
            (
            m
            +
            2
            )
          
        
        D
      
    
    {\displaystyle 2^{-(m+2)}D}
  
 is a Hausdorff sequentially complete subset of 
  
    
      
        (
        X
        ,
        τ
        )
        .
      
    
    {\displaystyle (X,\tau ).}
  
 
In particular, this is true for 
  
    
      
        
          x
          
            1
          
        
        +
        
          2
          
            −
            3
          
        
        D
      
    
    {\displaystyle x_{1}+2^{-3}D}
  
 so there exists some 
  
    
      
        x
        ∈
        
          x
          
            1
          
        
        +
        
          2
          
            −
            3
          
        
        D
      
    
    {\displaystyle x\in x_{1}+2^{-3}D}
  
 such that 
  
    
      
        
          x
          
            ∙
          
        
        →
        x
      
    
    {\displaystyle x_{\bullet }\to x}
  
 in 
  
    
      
        (
        X
        ,
        τ
        )
        .
      
    
    {\displaystyle (X,\tau ).}

Since 
 
 
 
 
 x
 
 n
 
 
 −
 
 x
 
 m
 
 
 ∈
 
 2
 
 −
 (
 m
 +
 2
 )
 
 
 D
 
 
 {\displaystyle x_{n}-x_{m}\in 2^{-(m+2)}D}
 
 for all 
 
 
 
 0
 <
 m
 <
 n
 ,
 
 
 {\displaystyle 0<m<n,}
 
 by fixing 
 
 
 
 m
 
 
 {\displaystyle m}
 
 and taking the limit (in 
 
 
 
 (
 X
 ,
 τ
 )
 
 
 {\displaystyle (X,\tau )}
 
) as 
 
 
 
 n
 →
 ∞
 ,
 
 
 {\displaystyle n\to \infty ,}
 
 it follows that 
 
 
 
 x
 −
 
 x
 
 m
 
 
 ∈
 
 2
 
 −
 (
 m
 +
 2
 )
 
 
 D
 
 
 {\displaystyle x-x_{m}\in 2^{-(m+2)}D}
 
 for each 
 
 
 
 m
 >
 0.
 
 
 {\displaystyle m>0.}
 
 
This implies that 
 
 
 
 p
 
 (
 
 x
 −
 
 x
 
 m
 
 
 
 )
 
 →
 0
 
 
 {\displaystyle p\left(x-x_{m}\right)\to 0}
 
 as 
 
 
 
 m
 →
 ∞
 ,
 
 
 {\displaystyle m\to \infty ,}
 
 which says exactly that 
 
 
 
 
 x
 
 ∙
 
 
 →
 x
 
 
 {\displaystyle x_{\bullet }\to x}
 
 in 
 
 
 
 
 (
 
 
 X
 
 D
 
 
 ,
 p
 
 )
 
 .
 
 
 {\displaystyle \left(X_{D},p\right).}
 
 
This shows that 
 
 
 
 
 (
 
 
 X
 
 D
 
 
 ,
 p
 
 )
 
 
 
 {\displaystyle \left(X_{D},p\right)}
 
 is complete.
†This assumption is allowed because 
 
 
 
 
 x
 
 ∙
 
 
 
 
 {\displaystyle x_{\bullet }}
 
 is a Cauchy sequence in a metric space (so the limits of all subsequences are equal) and a sequence in a metric space converges if and only if every subsequence has a sub-subsequence that converges.

Note that even if 
 
 
 
 D
 
 
 {\displaystyle D}
 
 is not a bounded and sequentially complete subset of any Hausdorff TVS, one might still be able to conclude that 
 
 
 
 
 (
 
 
 X
 
 D
 
 
 ,
 
 p
 
 D
 
 
 
 )
 
 
 
 {\displaystyle \left(X_{D},p_{D}\right)}
 
 is a Banach space by applying this theorem to some disk 
 
 
 
 K
 
 
 {\displaystyle K}
 
 satisfying

{
 
 x
 ∈
 span
 ⁡
 D
 :
 
 p
 
 D
 
 
 (
 x
 )
 <
 1
 
 }
 
 ⊆
 K
 ⊆
 
 {
 
 x
 ∈
 span
 ⁡
 D
 :
 
 p
 
 D
 
 
 (
 x
 )
 ≤
 1
 
 }
 
 
 
 {\displaystyle \left\{x\in \operatorname {span} D:p_{D}(x)<1\right\}\subseteq K\subseteq \left\{x\in \operatorname {span} D:p_{D}(x)\leq 1\right\}}

because 
  
    
      
        
          p
          
            D
          
        
        =
        
          p
          
            K
          
        
        .
      
    
    {\displaystyle p_{D}=p_{K}.}

The following are consequences of the above theorem:

<ul><li>A sequentially complete bounded disk in a Hausdorff TVS is a Banach disk.<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a></li>
<li>Any disk in a Hausdorff TVS that is complete and bounded (e.g. compact) is a Banach disk.<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a></li>
<li>The closed unit ball in a <a href="/facts/Fr%C3%A9chet_space/9gXkQu2u">Fréchet space</a> is sequentially complete and thus a Banach disk.<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a></li></ul>
Suppose that 
 
 
 
 D
 
 
 {\displaystyle D}
 
 is a bounded disk in a TVS 
 
 
 
 X
 .
 
 
 {\displaystyle X.}

<ul><li>If 
 
 
 
 L
 :
 X
 →
 Y
 
 
 {\displaystyle L:X\to Y}
 
 is a continuous linear map and 
 
 
 
 B
 ⊆
 X
 
 
 {\displaystyle B\subseteq X}
 
 is a Banach disk, then 
 
 
 
 L
 (
 B
 )
 
 
 {\displaystyle L(B)}
 
 is a Banach disk and 
 
 
 
 L
 
 
 
 |
 
 
 
 
 X
 
 B
 
 
 
 
 :
 
 X
 
 B
 
 
 →
 L
 
 (
 
 X
 
 B
 
 
 )
 
 
 
 {\displaystyle L{\big \vert }_{X_{B}}:X_{B}\to L\left(X_{B}\right)}
 
 induces an isometric TVS-isomorphism 
 
 
 
 
 Y
 
 L
 (
 B
 )
 
 
 ≅
 
 X
 
 B
 
 
 
 /
 
 
 (
 
 
 X
 
 B
 
 
 ∩
 ker
 ⁡
 L
 
 )
 
 .
 
 
 {\displaystyle Y_{L(B)}\cong X_{B}/\left(X_{B}\cap \operatorname {ker} L\right).}
 
</li></ul>
<h3>Properties of Banach disks</h3>
Let 
 
 
 
 X
 
 
 {\displaystyle X}
 
 be a TVS and let 
 
 
 
 D
 
 
 {\displaystyle D}
 
 be a bounded disk in 
 
 
 
 X
 .
 
 
 {\displaystyle X.}

If 
 
 
 
 D
 
 
 {\displaystyle D}
 
 is a bounded Banach disk in a Hausdorff locally convex space 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and if 
 
 
 
 T
 
 
 {\displaystyle T}
 
 is a barrel in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 then 
 
 
 
 T
 
 
 {\displaystyle T}
 
 <a href="/facts/Absorbing_set/GCtjsUuX">absorbs</a> 
 
 
 
 D
 
 
 {\displaystyle D}
 
 (that is, there is a number 
 
 
 
 r
 >
 0
 
 
 {\displaystyle r>0}
 
 such that 
 
 
 
 D
 ⊆
 r
 T
 .
 
 
 {\displaystyle D\subseteq rT.}
 
<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a>
If 
 
 
 
 U
 
 
 {\displaystyle U}
 
 is a convex balanced closed neighborhood of the origin in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 then the collection of all neighborhoods 
 
 
 
 r
 U
 ,
 
 
 {\displaystyle rU,}
 
 where 
 
 
 
 r
 >
 0
 
 
 {\displaystyle r>0}
 
 ranges over the positive real numbers, induces a topological vector space topology on 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 
 When 
 
 
 
 X
 
 
 {\displaystyle X}
 
 has this topology, it is denoted by 
 
 
 
 
 X
 
 U
 
 
 .
 
 
 {\displaystyle X_{U}.}
 
 Since this topology is not necessarily Hausdorff nor complete, the completion of the Hausdorff space 
 
 
 
 X
 
 /
 
 
 p
 
 U
 
 
 −
 1
 
 
 (
 0
 )
 
 
 {\displaystyle X/p_{U}^{-1}(0)}
 
 is denoted by 
 
 
 
 
 
 
 X
 
 U
 
 
 ¯
 
 
 
 
 {\displaystyle {\overline {X_{U}}}}
 
 so that 
 
 
 
 
 
 
 X
 
 U
 
 
 ¯
 
 
 
 
 {\displaystyle {\overline {X_{U}}}}
 
 is a complete Hausdorff space and 
 
 
 
 
 p
 
 U
 
 
 (
 x
 )
 :=
 
 inf
 
 x
 ∈
 r
 U
 ,
 r
 >
 0
 
 
 r
 
 
 {\displaystyle p_{U}(x):=\inf _{x\in rU,r>0}r}
 
 is a norm on this space making 
 
 
 
 
 
 
 X
 
 U
 
 
 ¯
 
 
 
 
 {\displaystyle {\overline {X_{U}}}}
 
 into a Banach space. The polar of 
 
 
 
 U
 ,
 
 
 {\displaystyle U,}
 
 
 
 
 
 
 U
 
 ∘
 
 
 ,
 
 
 {\displaystyle U^{\circ },}
 
 is a weakly compact bounded equicontinuous disk in 
 
 
 
 
 X
 
 ′
 
 
 
 
 {\displaystyle X^{\prime }}
 
 and so is infracomplete.
If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a <a href="/facts/Metrizable_topological_vector_space/6FUHfx6W">metrizable</a> <a href="/facts/Locally_convex_topological_vector_space/AGrPrayC">locally convex</a> TVS then for every <a href="/facts/Bounded_set_(topological_vector_space)/bTsWeU9g">bounded</a> subset 
 
 
 
 B
 
 
 {\displaystyle B}
 
 of 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 there exists a bounded <a href="/facts/Absolutely_convex_set/rJ5HLWkg">disk</a> 
 
 
 
 D
 
 
 {\displaystyle D}
 
 in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 such that 
 
 
 
 B
 ⊆
 
 X
 
 D
 
 
 ,
 
 
 {\displaystyle B\subseteq X_{D},}
 
 and both 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 
 X
 
 D
 
 
 
 
 {\displaystyle X_{D}}
 
 induce the same <a href="/facts/Subspace_topology/NjU6z4Ca">subspace topology</a> on 
 
 
 
 B
 .
 
 
 {\displaystyle B.}
 
<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a>

<h2 id="induced-by-a-radial-disk--quotient">Induced by a radial disk – quotient</h2>
Suppose that 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a topological vector space and 
 
 
 
 V
 
 
 {\displaystyle V}
 
 is a <a href="/facts/Convex_set/vdAuJRJl">convex</a> <a href="/facts/Balanced_set/Y2ar7E6g">balanced</a> and <a href="/facts/Radial_set/Sd027wad">radial</a> set. 
Then 
 
 
 
 
 {
 
 
 
 
 1
 n
 
 
 
 V
 :
 n
 =
 1
 ,
 2
 ,
 …
 
 }
 
 
 
 {\displaystyle \left\{{\tfrac {1}{n}}V:n=1,2,\ldots \right\}}
 
 is a neighborhood basis at the origin for some locally convex topology 
 
 
 
 
 τ
 
 V
 
 
 
 
 {\displaystyle \tau _{V}}
 
 on 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 
 
This TVS topology 
 
 
 
 
 τ
 
 V
 
 
 
 
 {\displaystyle \tau _{V}}
 
 is given by the <a href="/facts/Minkowski_functional/1qoFAjRs">Minkowski functional</a> formed by 
 
 
 
 V
 ,
 
 
 {\displaystyle V,}
 
 
 
 
 
 
 p
 
 V
 
 
 :
 X
 →
 
 R
 
 ,
 
 
 {\displaystyle p_{V}:X\to \mathbb {R} ,}
 
 which is a seminorm on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 defined by 
 
 
 
 
 p
 
 V
 
 
 (
 x
 )
 :=
 
 inf
 
 x
 ∈
 r
 V
 ,
 r
 >
 0
 
 
 r
 .
 
 
 {\displaystyle p_{V}(x):=\inf _{x\in rV,r>0}r.}
 
 
The topology 
 
 
 
 
 τ
 
 V
 
 
 
 
 {\displaystyle \tau _{V}}
 
 is Hausdorff if and only if 
 
 
 
 
 p
 
 V
 
 
 
 
 {\displaystyle p_{V}}
 
 is a norm, or equivalently, if and only if 
 
 
 
 X
 
 /
 
 
 p
 
 V
 
 
 −
 1
 
 
 (
 0
 )
 =
 {
 0
 }
 
 
 {\displaystyle X/p_{V}^{-1}(0)=\{0\}}
 
 or equivalently, for which it suffices that 
 
 
 
 V
 
 
 {\displaystyle V}
 
 be <a href="/facts/Von_Neumann_bounded/bTsWeU9g">bounded</a> in 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 
 
The topology 
 
 
 
 
 τ
 
 V
 
 
 
 
 {\displaystyle \tau _{V}}
 
 need not be Hausdorff but 
 
 
 
 X
 
 /
 
 
 p
 
 V
 
 
 −
 1
 
 
 (
 0
 )
 
 
 {\displaystyle X/p_{V}^{-1}(0)}
 
 is Hausdorff. 
A norm on 
 
 
 
 X
 
 /
 
 
 p
 
 V
 
 
 −
 1
 
 
 (
 0
 )
 
 
 {\displaystyle X/p_{V}^{-1}(0)}
 
 is given by 
 
 
 
 
 ‖
 
 x
 +
 X
 
 /
 
 
 p
 
 V
 
 
 −
 1
 
 
 (
 0
 )
 
 ‖
 
 :=
 
 p
 
 V
 
 
 (
 x
 )
 ,
 
 
 {\displaystyle \left\|x+X/p_{V}^{-1}(0)\right\|:=p_{V}(x),}
 
 where this value is in fact independent of the representative of the equivalence class 
 
 
 
 x
 +
 X
 
 /
 
 
 p
 
 V
 
 
 −
 1
 
 
 (
 0
 )
 
 
 {\displaystyle x+X/p_{V}^{-1}(0)}
 
 chosen. 
The normed space 
 
 
 
 
 (
 
 X
 
 /
 
 
 p
 
 V
 
 
 −
 1
 
 
 (
 0
 )
 ,
 ‖
 ⋅
 ‖
 
 )
 
 
 
 {\displaystyle \left(X/p_{V}^{-1}(0),\|\cdot \|\right)}
 
 is denoted by 
 
 
 
 
 X
 
 V
 
 
 
 
 {\displaystyle X_{V}}
 
 and its completion is denoted by 
 
 
 
 
 
 
 X
 
 V
 
 
 ¯
 
 
 .
 
 
 {\displaystyle {\overline {X_{V}}}.}

If in addition 
 
 
 
 V
 
 
 {\displaystyle V}
 
 is bounded in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 then the seminorm 
 
 
 
 
 p
 
 V
 
 
 :
 X
 →
 
 R
 
 
 
 {\displaystyle p_{V}:X\to \mathbb {R} }
 
 is a norm so in particular, 
 
 
 
 
 p
 
 V
 
 
 −
 1
 
 
 (
 0
 )
 =
 {
 0
 }
 .
 
 
 {\displaystyle p_{V}^{-1}(0)=\{0\}.}
 
 
In this case, we take 
 
 
 
 
 X
 
 V
 
 
 
 
 {\displaystyle X_{V}}
 
 to be the vector space 
 
 
 
 X
 
 
 {\displaystyle X}
 
 instead of 
 
 
 
 X
 
 /
 
 {
 0
 }
 
 
 {\displaystyle X/\{0\}}
 
 so that the notation 
 
 
 
 
 X
 
 V
 
 
 
 
 {\displaystyle X_{V}}
 
 is unambiguous (whether 
 
 
 
 
 X
 
 V
 
 
 
 
 {\displaystyle X_{V}}
 
 denotes the space induced by a radial disk or the space induced by a bounded disk).<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a>
The <a href="/facts/Quotient_topology/muE0qQVC">quotient topology</a> 
 
 
 
 
 τ
 
 Q
 
 
 
 
 {\displaystyle \tau _{Q}}
 
 on 
 
 
 
 X
 
 /
 
 
 p
 
 V
 
 
 −
 1
 
 
 (
 0
 )
 
 
 {\displaystyle X/p_{V}^{-1}(0)}
 
 (inherited from 
 
 
 
 X
 
 
 {\displaystyle X}
 
's original topology) is finer (in general, strictly finer) than the norm topology.

<h3>Canonical maps</h3>
The canonical map is the <a href="/facts/Quotient_map/muE0qQVC">quotient map</a> 
 
 
 
 
 q
 
 V
 
 
 :
 X
 →
 
 X
 
 V
 
 
 =
 X
 
 /
 
 
 p
 
 V
 
 
 −
 1
 
 
 (
 0
 )
 ,
 
 
 {\displaystyle q_{V}:X\to X_{V}=X/p_{V}^{-1}(0),}
 
 which is continuous when 
 
 
 
 
 X
 
 V
 
 
 
 
 {\displaystyle X_{V}}
 
 has either the norm topology or the quotient topology.<a class="footnote-ref" id="fnref:20" href="#fn:20">20</a>
If 
 
 
 
 U
 
 
 {\displaystyle U}
 
 and 
 
 
 
 V
 
 
 {\displaystyle V}
 
 are radial disks such that 
 
 
 
 U
 ⊆
 V
 
 
 {\displaystyle U\subseteq V}
 
 then 
 
 
 
 
 p
 
 U
 
 
 −
 1
 
 
 (
 0
 )
 ⊆
 
 p
 
 V
 
 
 −
 1
 
 
 (
 0
 )
 
 
 {\displaystyle p_{U}^{-1}(0)\subseteq p_{V}^{-1}(0)}
 
 so there is a continuous linear surjective canonical map 
 
 
 
 
 q
 
 V
 ,
 U
 
 
 :
 X
 
 /
 
 
 p
 
 U
 
 
 −
 1
 
 
 (
 0
 )
 →
 X
 
 /
 
 
 p
 
 V
 
 
 −
 1
 
 
 (
 0
 )
 =
 
 X
 
 V
 
 
 
 
 {\displaystyle q_{V,U}:X/p_{U}^{-1}(0)\to X/p_{V}^{-1}(0)=X_{V}}
 
 defined by sending

x
 +
 
 p
 
 U
 
 
 −
 1
 
 
 (
 0
 )
 ∈
 
 X
 
 U
 
 
 =
 X
 
 /
 
 
 p
 
 U
 
 
 −
 1
 
 
 (
 0
 )
 
 
 {\displaystyle x+p_{U}^{-1}(0)\in X_{U}=X/p_{U}^{-1}(0)}
 
 to the equivalence class 
 
 
 
 x
 +
 
 p
 
 V
 
 
 −
 1
 
 
 (
 0
 )
 ,
 
 
 {\displaystyle x+p_{V}^{-1}(0),}
 
 where one may verify that the definition does not depend on the representative of the equivalence class 
 
 
 
 x
 +
 
 p
 
 U
 
 
 −
 1
 
 
 (
 0
 )
 
 
 {\displaystyle x+p_{U}^{-1}(0)}
 
 that is chosen.<a class="footnote-ref" id="fnref:21" href="#fn:21">21</a> 
This canonical map has norm 
 
 
 
 
 ≤
 1
 
 
 {\displaystyle \,\leq 1}
 
<a class="footnote-ref" id="fnref:22" href="#fn:22">22</a> and it has a unique continuous linear canonical extension to 
 
 
 
 
 
 
 X
 
 U
 
 
 ¯
 
 
 
 
 {\displaystyle {\overline {X_{U}}}}
 
 that is denoted by 
 
 
 
 
 
 
 g
 
 V
 ,
 U
 
 
 ¯
 
 
 :
 
 
 
 X
 
 U
 
 
 ¯
 
 
 →
 
 
 
 X
 
 V
 
 
 ¯
 
 
 .
 
 
 {\displaystyle {\overline {g_{V,U}}}:{\overline {X_{U}}}\to {\overline {X_{V}}}.}

Suppose that in addition 
 
 
 
 B
 ≠
 ∅
 
 
 {\displaystyle B\neq \varnothing }
 
 and 
 
 
 
 C
 
 
 {\displaystyle C}
 
 are bounded disks in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 with 
 
 
 
 B
 ⊆
 C
 
 
 {\displaystyle B\subseteq C}
 
 so that 
 
 
 
 
 X
 
 B
 
 
 ⊆
 
 X
 
 C
 
 
 
 
 {\displaystyle X_{B}\subseteq X_{C}}
 
 and the inclusion 
 
 
 
 
 In
 
 B
 
 
 C
 
 
 :
 
 X
 
 B
 
 
 →
 
 X
 
 C
 
 
 
 
 {\displaystyle \operatorname {In} _{B}^{C}:X_{B}\to X_{C}}
 
 is a continuous linear map. 
Let 
 
 
 
 
 In
 
 B
 
 
 :
 
 X
 
 B
 
 
 →
 X
 ,
 
 
 {\displaystyle \operatorname {In} _{B}:X_{B}\to X,}
 
 
 
 
 
 
 In
 
 C
 
 
 :
 
 X
 
 C
 
 
 →
 X
 ,
 
 
 {\displaystyle \operatorname {In} _{C}:X_{C}\to X,}
 
 and 
 
 
 
 
 In
 
 B
 
 
 C
 
 
 :
 
 X
 
 B
 
 
 →
 
 X
 
 C
 
 
 
 
 {\displaystyle \operatorname {In} _{B}^{C}:X_{B}\to X_{C}}
 
 be the canonical maps. 
Then 
 
 
 
 
 In
 
 C
 
 
 =
 
 In
 
 B
 
 
 C
 
 
 ∘
 
 In
 
 C
 
 
 :
 
 X
 
 B
 
 
 →
 
 X
 
 C
 
 
 
 
 {\displaystyle \operatorname {In} _{C}=\operatorname {In} _{B}^{C}\circ \operatorname {In} _{C}:X_{B}\to X_{C}}
 
 and 
 
 
 
 
 q
 
 V
 
 
 =
 
 q
 
 V
 ,
 U
 
 
 ∘
 
 q
 
 U
 
 
 .
 
 
 {\displaystyle q_{V}=q_{V,U}\circ q_{U}.}
 
<a class="footnote-ref" id="fnref:23" href="#fn:23">23</a>

<h2 id="induced-by-a-bounded-radial-disk">Induced by a bounded radial disk</h2>
Suppose that 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is a bounded radial disk. 
Since 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is a bounded disk, if 
 
 
 
 D
 :=
 S
 
 
 {\displaystyle D:=S}
 
 then we may create the auxiliary normed space 
 
 
 
 
 X
 
 D
 
 
 =
 span
 ⁡
 D
 
 
 {\displaystyle X_{D}=\operatorname {span} D}
 
 with norm 
 
 
 
 
 p
 
 D
 
 
 (
 x
 )
 :=
 
 inf
 
 x
 ∈
 r
 D
 ,
 r
 >
 0
 
 
 r
 
 
 {\displaystyle p_{D}(x):=\inf _{x\in rD,r>0}r}
 
; since 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is radial, 
 
 
 
 
 X
 
 S
 
 
 =
 X
 .
 
 
 {\displaystyle X_{S}=X.}
 
 
Since 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is a radial disk, if 
 
 
 
 V
 :=
 S
 
 
 {\displaystyle V:=S}
 
 then we may create the auxiliary seminormed space 
 
 
 
 X
 
 /
 
 
 p
 
 V
 
 
 −
 1
 
 
 (
 0
 )
 
 
 {\displaystyle X/p_{V}^{-1}(0)}
 
 with the seminorm 
 
 
 
 
 p
 
 V
 
 
 (
 x
 )
 :=
 
 inf
 
 x
 ∈
 r
 V
 ,
 r
 >
 0
 
 
 r
 
 
 {\displaystyle p_{V}(x):=\inf _{x\in rV,r>0}r}
 
; because 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is bounded, this seminorm is a norm and 
 
 
 
 
 p
 
 V
 
 
 −
 1
 
 
 (
 0
 )
 =
 {
 0
 }
 
 
 {\displaystyle p_{V}^{-1}(0)=\{0\}}
 
 so 
 
 
 
 X
 
 /
 
 
 p
 
 V
 
 
 −
 1
 
 
 (
 0
 )
 =
 X
 
 /
 
 {
 0
 }
 =
 X
 .
 
 
 {\displaystyle X/p_{V}^{-1}(0)=X/\{0\}=X.}
 
 
Thus, in this case the two auxiliary normed spaces produced by these two different methods result in the same normed space.

<h2 id="duality">Duality</h2>
Suppose that 
 
 
 
 H
 
 
 {\displaystyle H}
 
 is a weakly closed equicontinuous disk in 
 
 
 
 
 X
 
 ′
 
 
 
 
 {\displaystyle X^{\prime }}
 
 (this implies that 
 
 
 
 H
 
 
 {\displaystyle H}
 
 is weakly compact) and let

U
        :=
        
          H
          
            ∘
          
        
        =
        {
        x
        ∈
        X
        :
        
          |
        
        h
        (
        x
        )
        
          |
        
        ≤
        1
        
           for all 
        
        h
        ∈
        H
        }
      
    
    {\displaystyle U:=H^{\circ }=\{x\in X:|h(x)|\leq 1{\text{ for all }}h\in H\}}

be the <a href="/facts/Polar_set/JY45pfbt">polar</a> of 
 
 
 
 H
 .
 
 
 {\displaystyle H.}
 
 
Because 
 
 
 
 
 U
 
 ∘
 
 
 =
 
 H
 
 ∘
 ∘
 
 
 =
 H
 
 
 {\displaystyle U^{\circ }=H^{\circ \circ }=H}
 
 by the <a href="/facts/Bipolar_theorem/MtShefAp">bipolar theorem</a>, it follows that a continuous linear functional 
 
 
 
 f
 
 
 {\displaystyle f}
 
 belongs to 
 
 
 
 
 X
 
 H
 
 
 ′
 
 
 =
 span
 ⁡
 H
 
 
 {\displaystyle X_{H}^{\prime }=\operatorname {span} H}
 
 if and only if 
 
 
 
 f
 
 
 {\displaystyle f}
 
 belongs to the continuous dual space of 
 
 
 
 
 (
 
 X
 ,
 
 p
 
 U
 
 
 
 )
 
 ,
 
 
 {\displaystyle \left(X,p_{U}\right),}
 
 where 
 
 
 
 
 p
 
 U
 
 
 
 
 {\displaystyle p_{U}}
 
 is the <a href="/facts/Minkowski_functional/1qoFAjRs">Minkowski functional</a> of 
 
 
 
 U
 
 
 {\displaystyle U}
 
 defined by 
 
 
 
 
 p
 
 U
 
 
 (
 x
 )
 :=
 
 inf
 
 x
 ∈
 r
 U
 ,
 r
 >
 0
 
 
 r
 .
 
 
 {\displaystyle p_{U}(x):=\inf _{x\in rU,r>0}r.}
 
<a class="footnote-ref" id="fnref:24" href="#fn:24">24</a>

<h2 id="related-concepts">Related concepts</h2>
A disk in a TVS is called <a href="/facts/Infrabornivorous/xwVBUY0M">infrabornivorous</a><a class="footnote-ref" id="fnref:25" href="#fn:25">25</a> if it <a href="/facts/Absorbing_set/GCtjsUuX">absorbs</a> all Banach disks.
A linear map between two TVSs is called <a href="/facts/Infrabounded_map/xwVBUY0M">infrabounded</a><a class="footnote-ref" id="fnref:26" href="#fn:26">26</a> if it maps Banach disks to bounded disks.

<h3>Fast convergence</h3>
A sequence 
 
 
 
 
 x
 
 ∙
 
 
 =
 
 
 (
 
 x
 
 i
 
 
 )
 
 
 i
 =
 1
 
 
 ∞
 
 
 
 
 {\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }}
 
 in a TVS 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is said to be fast convergent<a class="footnote-ref" id="fnref:27" href="#fn:27">27</a> to a point 
 
 
 
 x
 ∈
 X
 
 
 {\displaystyle x\in X}
 
 if there exists a Banach disk 
 
 
 
 D
 
 
 {\displaystyle D}
 
 such that both 
 
 
 
 x
 
 
 {\displaystyle x}
 
 and the sequence is (eventually) contained in 
 
 
 
 span
 ⁡
 D
 
 
 {\displaystyle \operatorname {span} D}
 
 and 
 
 
 
 
 x
 
 ∙
 
 
 →
 x
 
 
 {\displaystyle x_{\bullet }\to x}
 
 in 
 
 
 
 
 (
 
 
 X
 
 D
 
 
 ,
 
 p
 
 D
 
 
 
 )
 
 .
 
 
 {\displaystyle \left(X_{D},p_{D}\right).}

Every fast convergent sequence is <a href="/facts/Mackey_convergent_sequence/kpl5B2Ez">Mackey convergent</a>.<a class="footnote-ref" id="fnref:28" href="#fn:28">28</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Bornological_space/kpl5B2Ez">Bornological space</a> – Space where bounded operators are continuous</li>
<li><a href="/facts/Injective_tensor_product/53zV1QtL">Injective tensor product</a></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/Nuclear_operator/kxnx5bIJ">Nuclear operator</a> – Linear operator related to topological vector spaces</li>
<li><a href="/facts/Nuclear_space/4FLPWdfp">Nuclear space</a> – A generalization of finite-dimensional Euclidean spaces different from Hilbert spaces</li>
<li><a href="/facts/Initial_topology/bNRdJM81">Initial topology</a> – Coarsest topology making certain functions continuous</li>
<li><a href="/facts/Projective_tensor_product/lv5X7Ps7">Projective tensor product</a> – tensor product defined on two topological vector spacesPages displaying wikidata descriptions as a fallback</li>
<li><a href="/facts/Schwartz_topological_vector_space/c559KL0o">Schwartz topological vector space</a> – topological vector space whose neighborhoods of the origin have a property similar to the definition of totally bounded subsetsPages displaying wikidata descriptions as a fallback</li>
<li><a href="/facts/Tensor_product_of_Hilbert_spaces/GSavLJfU">Tensor product of Hilbert spaces</a> – Tensor product space endowed with a special inner product</li>
<li><a href="/facts/Topological_tensor_product/eDICDJ2Y">Topological tensor product</a> – Tensor product constructions for topological vector spaces</li>
<li><a href="/facts/Ultrabornological_space/DLUE8S0K">Ultrabornological space</a></li></ul>
<h2 id="notes">Notes</h2>

<h2 id="bibliography">Bibliography</h2>
<ul><li>Burzyk, Józef; Gilsdorf, Thomas E. (1995). <a href="https://downloads.hindawi.com/journals/ijmms/1995/138625.pdf">"Some remarks about Mackey convergence"</a> (PDF). International Journal of Mathematics and Mathematical Sciences. 18 (4). Hindawi Limited: 659–664. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1155%2Fs0161171295000846">10.1155/s0161171295000846</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0161-1712">0161-1712</a>.</li>
<li>Diestel, Joe (2008). The Metric Theory of Tensor Products: Grothendieck's Résumé Revisited. Vol. 16. Providence, R.I.: <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9781470424831. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/185095773">185095773</a>.</li>
<li>Dubinsky, Ed (1979). The Structure of Nuclear Fréchet Spaces. <a href="/facts/Lecture_Notes_in_Mathematics/5DmSFMhJ">Lecture Notes in Mathematics</a>. Vol. 720. Berlin New York: <a href="/facts/Springer_Publishing/JfRev59b">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-09504-0. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/5126156">5126156</a>.</li>
<li><a href="/facts/Alexander_Grothendieck/jLQGPY7M">Grothendieck, Alexander</a> (1955). "Produits Tensoriels Topologiques et Espaces Nucléaires" [Topological Tensor Products and Nuclear Spaces]. Memoirs of the American Mathematical Society Series (in French). 16. Providence: American Mathematical Society. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-1216-7. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0075539">0075539</a>. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/1315788">1315788</a>.</li>
<li><a href="/facts/Henri_Hogbe-Nlend/cwWrdFXu">Hogbe-Nlend, Henri</a> (1977). Bornologies and Functional Analysis: Introductory Course on the Theory of Duality Topology-Bornology and its use in Functional Analysis. North-Holland Mathematics Studies. Vol. 26. Amsterdam New York New York: North Holland. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-08-087137-0. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0500064">0500064</a>. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/316549583">316549583</a>.</li>
<li><a href="/facts/Henri_Hogbe-Nlend/cwWrdFXu">Hogbe-Nlend, Henri</a>; Moscatelli, V. B. (1981). Nuclear and Conuclear Spaces: Introductory Course on Nuclear and Conuclear Spaces in the Light of the Duality "topology-bornology". North-Holland Mathematics Studies. Vol. 52. Amsterdam New York New York: North Holland. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-08-087163-9. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/316564345">316564345</a>.</li>
<li>Husain, Taqdir; Khaleelulla, S. M. (1978). Barrelledness in Topological and Ordered Vector Spaces. <a href="/facts/Lecture_Notes_in_Mathematics/5DmSFMhJ">Lecture Notes in Mathematics</a>. Vol. 692. Berlin, New York, Heidelberg: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-09096-0. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/4493665">4493665</a>.</li>
<li>Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. <a href="/facts/Lecture_Notes_in_Mathematics/5DmSFMhJ">Lecture Notes in Mathematics</a>. Vol. 936. Berlin, Heidelberg, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-11565-6. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/8588370">8588370</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>Pietsch, Albrecht (1979). Nuclear Locally Convex Spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 66 (Second ed.). Berlin, New York: Springer-Verlag. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-05644-9. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/539541">539541</a>.</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>Ryan, Raymond A. (2002). Introduction to Tensor Products of Banach Spaces. Springer Monographs in Mathematics. London New York: <a href="/facts/Springer_Publishing/JfRev59b">Springer</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-85233-437-6. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/48092184">48092184</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/Fran%C3%A7ois_Tr%C3%A8ves/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>Wong, Yau-Chuen (1979). Schwartz Spaces, Nuclear Spaces, and Tensor Products. <a href="/facts/Lecture_Notes_in_Mathematics/5DmSFMhJ">Lecture Notes in Mathematics</a>. Vol. 726. Berlin New York: <a href="/facts/Springer_Publishing/JfRev59b">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-09513-2. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/5126158">5126158</a>.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="https://ncatlab.org/nlab/show/nuclear+space">Nuclear space at ncatlab</a></li></ul>

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

<ol>
<li id="fn:1">Schaefer & Wolff 1999, p. 97. - 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:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Schaefer & Wolff 1999, p. 169. - 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:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">This is the smallest vector space containing 
 
 
 
 ∅
 .
 
 
 {\displaystyle \varnothing .}
 
 Alternatively, if 
 
 
 
 D
 =
 ∅
 
 
 {\displaystyle D=\varnothing }
 
 then 
 
 
 
 D
 
 
 {\displaystyle D}
 
 may instead be replaced with 
 
 
 
 {
 0
 }
 .
 
 
 {\displaystyle \{0\}.}
 <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Schaefer & Wolff 1999, p. 97. - 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:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Trèves 2006, p. 370. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. <a href="https://search.worldcat.org/oclc/853623322" target="_blank">https://search.worldcat.org/oclc/853623322</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Trèves 2006, pp. 370–373. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. <a href="https://search.worldcat.org/oclc/853623322" target="_blank">https://search.worldcat.org/oclc/853623322</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Narici & Beckenstein 2011, pp. 441–457. - 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:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Schaefer & Wolff 1999, p. 97. - 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:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">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:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Narici & Beckenstein 2011, pp. 441–457. - 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:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Narici & Beckenstein 2011, pp. 441–457. - 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:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Assume WLOG that 
 
 
 
 X
 =
 span
 ⁡
 D
 .
 
 
 {\displaystyle X=\operatorname {span} D.}
 
 Since 
 
 
 
 D
 
 
 {\displaystyle D}
 
 is closed in 
 
 
 
 (
 X
 ,
 τ
 )
 ,
 
 
 {\displaystyle (X,\tau ),}
 
 it is also closed in 
 
 
 
 
 (
 
 
 X
 
 D
 
 
 ,
 
 p
 
 D
 
 
 
 )
 
 
 
 {\displaystyle \left(X_{D},p_{D}\right)}
 
 and since the seminorm 
 
 
 
 
 p
 
 D
 
 
 
 
 {\displaystyle p_{D}}
 
 is the Minkowski functional of 
 
 
 
 D
 ,
 
 
 {\displaystyle D,}
 
 which is continuous on 
 
 
 
 
 (
 
 
 X
 
 D
 
 
 ,
 
 p
 
 D
 
 
 
 )
 
 ,
 
 
 {\displaystyle \left(X_{D},p_{D}\right),}
 
 it follows Narici & Beckenstein (2011, pp. 119–120) that 
 
 
 
 D
 
 
 {\displaystyle D}
 
 is the closed unit ball in 
 
 
 
 
 (
 
 
 X
 
 D
 
 
 ,
 p
 
 )
 
 .
 
 
 {\displaystyle \left(X_{D},p\right).}
 <a href="/wiki/Minkowski_functional" target="_blank">/wiki/Minkowski_functional</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Narici & Beckenstein 2011, pp. 441–442. - 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">Narici & Beckenstein 2011, pp. 441–457. - 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:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Trèves 2006, pp. 370–371. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. <a href="https://search.worldcat.org/oclc/853623322" target="_blank">https://search.worldcat.org/oclc/853623322</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Narici & Beckenstein 2011, pp. 441–457. - 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:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Trèves 2006, pp. 370–373. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. <a href="https://search.worldcat.org/oclc/853623322" target="_blank">https://search.worldcat.org/oclc/853623322</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">Narici & Beckenstein 2011, pp. 441–457. - 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:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Schaefer & Wolff 1999, p. 97. - 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:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">Schaefer & Wolff 1999, p. 97. - 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:20" class="footnote-back-ref">↩</a></li>
<li id="fn:21">Schaefer & Wolff 1999, p. 97. - 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:21" class="footnote-back-ref">↩</a></li>
<li id="fn:22">Schaefer & Wolff 1999, p. 97. - 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:22" class="footnote-back-ref">↩</a></li>
<li id="fn:23">Schaefer & Wolff 1999, p. 97. - 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:23" class="footnote-back-ref">↩</a></li>
<li id="fn:24">Trèves 2006, p. 477. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. <a href="https://search.worldcat.org/oclc/853623322" target="_blank">https://search.worldcat.org/oclc/853623322</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></li>
<li id="fn:25">Narici & Beckenstein 2011, pp. 441–457. - 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:25" class="footnote-back-ref">↩</a></li>
<li id="fn:26">Narici & Beckenstein 2011, pp. 441–457. - 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:26" class="footnote-back-ref">↩</a></li>
<li id="fn:27">Narici & Beckenstein 2011, pp. 441–457. - 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">Narici & Beckenstein 2011, pp. 441–457. - 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:28" class="footnote-back-ref">↩</a></li>
</ol>

Auxiliary normed space open-in-new

Auxiliary normed space