Complemented subspace

<h2 id="preliminaries-definitions-and-notation">Preliminaries: definitions and notation</h2>
If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a vector space and 
 
 
 
 M
 
 
 {\displaystyle M}
 
 and 
 
 
 
 N
 
 
 {\displaystyle N}
 
 are <a href="/facts/Vector_subspace/2wtKTgHL">vector subspaces</a> of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 then there is a well-defined addition map

S
                :
                
              
              
              
                M
                ×
                N
              
              
              
                
                →
                
              
              
                X
              
            
            
              
              
              
                (
                m
                ,
                n
                )
              
              
              
                
                ↦
                
              
              
                m
                +
                n
              
            
          
        
      
    
    {\displaystyle {\begin{alignedat}{4}S:\;&&M\times N&&\;\to \;&X\\&&(m,n)&&\;\mapsto \;&m+n\\\end{alignedat}}}

The map 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is a <a href="/facts/Morphism/Kdpuyz3S">morphism</a> in the <a href="/facts/Category_of_vector_spaces/AnnF9JKf">category of vector spaces</a> — that is to say, <a href="/facts/Linear_map/Q1PBKNVV">linear</a>. 

<h3>Algebraic direct sum</h3>
Main articles: <a href="/facts/Direct_sum/x3ljVsP6">Direct sum</a> and <a href="/facts/Direct_sum_of_modules/Y6PBJnjg">Direct sum of modules</a>
The vector space 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is said to be the algebraic <a href="/facts/Direct_sum/x3ljVsP6">direct sum</a> (or direct sum in the category of vector spaces) 
 
 
 
 M
 ⊕
 N
 
 
 {\displaystyle M\oplus N}
 
 when any of the following equivalent conditions are satisfied:

<ol><li>The addition map 
 
 
 
 S
 :
 M
 ×
 N
 →
 X
 
 
 {\displaystyle S:M\times N\to X}
 
 is a <a href="/facts/Vector_space_isomorphism/Q1PBKNVV">vector space isomorphism</a>.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a><a class="footnote-ref" id="fnref:2" href="#fn:2">2</a></li>
<li>The addition map is bijective.</li>
<li>
 
 
 
 M
 ∩
 N
 =
 {
 0
 }
 
 
 {\displaystyle M\cap N=\{0\}}
 
 and 
 
 
 
 M
 +
 N
 =
 X
 
 
 {\displaystyle M+N=X}
 
; in this case 
 
 
 
 N
 
 
 {\displaystyle N}
 
 is called an algebraic complement or supplement to 
 
 
 
 M
 
 
 {\displaystyle M}
 
 in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and the two subspaces are said to be complementary or supplementary.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a><a class="footnote-ref" id="fnref:4" href="#fn:4">4</a></li></ol>
When these conditions hold, the inverse 
 
 
 
 
 S
 
 −
 1
 
 
 :
 X
 →
 M
 ×
 N
 
 
 {\displaystyle S^{-1}:X\to M\times N}
 
 is well-defined and can be written in terms of coordinates as
 
 
 
 
 S
 
 −
 1
 
 
 =
 
 (
 
 
 P
 
 M
 
 
 ,
 
 P
 
 N
 
 
 
 )
 
 
 .
 
 
 
 {\displaystyle S^{-1}=\left(P_{M},P_{N}\right){\text{.}}}

The first coordinate 
 
 
 
 
 P
 
 M
 
 
 :
 X
 →
 M
 
 
 {\displaystyle P_{M}:X\to M}
 
 is called the canonical projection of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 onto 
 
 
 
 M
 
 
 {\displaystyle M}
 
; likewise the second coordinate is the canonical projection onto 
 
 
 
 N
 .
 
 
 {\displaystyle N.}
 
<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> 
Equivalently, 
 
 
 
 
 P
 
 M
 
 
 (
 x
 )
 
 
 {\displaystyle P_{M}(x)}
 
 and 
 
 
 
 
 P
 
 N
 
 
 (
 x
 )
 
 
 {\displaystyle P_{N}(x)}
 
 are the unique vectors in 
 
 
 
 M
 
 
 {\displaystyle M}
 
 and 
 
 
 
 N
 ,
 
 
 {\displaystyle N,}
 
 respectively, that satisfy

x
        =
        
          P
          
            M
          
        
        (
        x
        )
        +
        
          P
          
            N
          
        
        (
        x
        )
        
          .
        
      
    
    {\displaystyle x=P_{M}(x)+P_{N}(x){\text{.}}}

As maps, 
 
 
 
 
 P
 
 M
 
 
 +
 
 P
 
 N
 
 
 =
 
 Id
 
 X
 
 
 ,
 
 ker
 ⁡
 
 P
 
 M
 
 
 =
 N
 ,
 
 
  and 
 
 
 ker
 ⁡
 
 P
 
 N
 
 
 =
 M
 
 
 {\displaystyle P_{M}+P_{N}=\operatorname {Id} _{X},\qquad \ker P_{M}=N,\qquad {\text{ and }}\qquad \ker P_{N}=M}
 
 where 
 
 
 
 
 Id
 
 X
 
 
 
 
 {\displaystyle \operatorname {Id} _{X}}
 
 denotes the <a href="/facts/Identity_map/9AtsP0LC">identity map</a> on 
 
 
 
 X
 
 
 {\displaystyle X}
 
.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

<h2 id="motivation">Motivation</h2>
See also: <a href="/facts/Coproduct/0bJ8SCdu">Coproduct</a>, <a href="/facts/Direct_sum/x3ljVsP6">Direct sum § Direct sum in categories</a>, and <a href="/facts/Direct_sum_of_topological_groups/rezTtyTf">Direct sum of topological groups</a>
Suppose that the vector space 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is the algebraic direct sum of 
 
 
 
 M
 ⊕
 N
 
 
 {\displaystyle M\oplus N}
 
. In the category of vector spaces, finite <a href="/facts/Product_(category_theory)/evFecz0t">products</a> and <a href="/facts/Coproduct/0bJ8SCdu">coproducts</a> coincide: algebraically, 
 
 
 
 M
 ⊕
 N
 
 
 {\displaystyle M\oplus N}
 
 and 
 
 
 
 M
 ×
 N
 
 
 {\displaystyle M\times N}
 
 are indistinguishable. Given a problem involving elements of 
 
 
 
 X
 
 
 {\displaystyle X}
 
, one can break the elements down into their components in 
 
 
 
 M
 
 
 {\displaystyle M}
 
 and 
 
 
 
 N
 
 
 {\displaystyle N}
 
, because the projection maps defined above act as inverses to the natural inclusion of 
 
 
 
 M
 
 
 {\displaystyle M}
 
 and 
 
 
 
 N
 
 
 {\displaystyle N}
 
 into 
 
 
 
 X
 
 
 {\displaystyle X}
 
. Then one can solve the problem in the vector subspaces and recombine to form an element of 
 
 
 
 X
 
 
 {\displaystyle X}
 
. 
In the category of <a href="/facts/Topological_vector_space/XRex1ChN">topological vector spaces</a>, that algebraic decomposition becomes less useful. The definition of a topological vector space requires the addition map 
 
 
 
 S
 
 
 {\displaystyle S}
 
 to be continuous; its inverse 
 
 
 
 
 S
 
 −
 1
 
 
 :
 X
 →
 M
 ×
 N
 
 
 {\displaystyle S^{-1}:X\to M\times N}
 
 may not be.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> The <a href="/facts/Coproduct/0bJ8SCdu">categorical definition of direct sum</a>, however, requires 
 
 
 
 
 P
 
 M
 
 
 
 
 {\displaystyle P_{M}}
 
 and 
 
 
 
 
 P
 
 N
 
 
 
 
 {\displaystyle P_{N}}
 
 to be morphisms — that is, continuous linear maps. 
The space 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is the topological direct sum of 
 
 
 
 M
 
 
 {\displaystyle M}
 
 and 
 
 
 
 N
 
 
 {\displaystyle N}
 
 if (and only if) any of the following equivalent conditions hold:

<ol><li>The addition map 
 
 
 
 S
 :
 M
 ×
 N
 →
 X
 
 
 {\displaystyle S:M\times N\to X}
 
 is a <a href="/facts/TVS-isomorphism/XRex1ChN">TVS-isomorphism</a> (that is, a surjective <a href="/facts/Linear_map/Q1PBKNVV">linear</a> <a href="/facts/Homeomorphism/jMfWg3Mn">homeomorphism</a>).<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a></li>
<li>
 
 
 
 X
 
 
 {\displaystyle X}
 
 is the algebraic direct sum of 
 
 
 
 M
 
 
 {\displaystyle M}
 
 and 
 
 
 
 N
 
 
 {\displaystyle N}
 
 and also any of the following equivalent conditions: <ol><li>The inverse of the addition map 
 
 
 
 
 S
 
 −
 1
 
 
 :
 X
 →
 M
 ×
 N
 
 
 {\displaystyle S^{-1}:X\to M\times N}
 
 is continuous.</li><li>Both canonical projections 
 
 
 
 
 P
 
 M
 
 
 :
 X
 →
 M
 
 
 {\displaystyle P_{M}:X\to M}
 
 and 
 
 
 
 
 P
 
 N
 
 
 :
 X
 →
 N
 
 
 {\displaystyle P_{N}:X\to N}
 
 are continuous.</li><li>At least one of the canonical projections 
 
 
 
 
 P
 
 M
 
 
 
 
 {\displaystyle P_{M}}
 
 and 
 
 
 
 
 P
 
 N
 
 
 
 
 {\displaystyle P_{N}}
 
 is continuous.</li><li>The canonical <a href="/facts/Quotient_map/muE0qQVC">quotient map</a> 
 
 
 
 p
 :
 N
 →
 X
 
 /
 
 M
 ;
 p
 (
 n
 )
 =
 n
 +
 M
 
 
 {\displaystyle p:N\to X/M;p(n)=n+M}
 
 is an isomorphism of topological vector spaces (i.e. a linear homeomorphism).<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a></li></ol></li>
<li>
 
 
 
 X
 
 
 {\displaystyle X}
 
 is the <a href="/facts/Coproduct/0bJ8SCdu">direct sum</a> of 
 
 
 
 M
 
 
 {\displaystyle M}
 
 and 
 
 
 
 N
 
 
 {\displaystyle N}
 
 in the category of topological vector spaces.</li>
<li>The map 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is <a href="/facts/Bijective/j4gTuTmW">bijective</a> and <a href="/facts/Open_map/abHUKX7l">open</a>.</li>
<li>When considered as <a href="/facts/Additive_group/djWVpp9D">additive</a> <a href="/facts/Topological_group/cAgNMMRU">topological groups</a>, 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is the <a href="/facts/Direct_sum_of_topological_groups/rezTtyTf">topological direct sum of the subgroups</a> 
 
 
 
 M
 
 
 {\displaystyle M}
 
 and 
 
 
 
 N
 .
 
 
 {\displaystyle N.}
 
</li></ol>
The topological direct sum is also written 
 
 
 
 X
 =
 M
 ⊕
 N
 
 
 {\displaystyle X=M\oplus N}
 
; whether the sum is in the topological or algebraic sense is usually clarified through <a href="/facts/Context_(language_use)/eJM44o7x">context</a>.

<h2 id="definition">Definition</h2>
Every topological direct sum is an algebraic direct sum 
 
 
 
 X
 =
 M
 ⊕
 N
 
 
 {\displaystyle X=M\oplus N}
 
; the converse is not guaranteed. Even if both 
 
 
 
 M
 
 
 {\displaystyle M}
 
 and 
 
 
 
 N
 
 
 {\displaystyle N}
 
 are closed in 
 
 
 
 X
 
 
 {\displaystyle X}
 
, 
 
 
 
 
 S
 
 −
 1
 
 
 
 
 {\displaystyle S^{-1}}
 
 may still fail to be continuous. 
 
 
 
 N
 
 
 {\displaystyle N}
 
 is a (topological) complement or supplement to 
 
 
 
 M
 
 
 {\displaystyle M}
 
 if it avoids that pathology — that is, if, topologically, 
 
 
 
 X
 =
 M
 ⊕
 N
 
 
 {\displaystyle X=M\oplus N}
 
. (Then 
 
 
 
 M
 
 
 {\displaystyle M}
 
 is likewise complementary to 
 
 
 
 N
 
 
 {\displaystyle N}
 
.)<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> Condition 2(d) above implies that any topological complement of 
 
 
 
 M
 
 
 {\displaystyle M}
 
 is isomorphic, as a topological vector space, to the <a href="/facts/Quotient_vector_space/WmQ3lqdI">quotient vector space</a> 
 
 
 
 X
 
 /
 
 M
 
 
 {\displaystyle X/M}
 
.

 
 
 
 M
 
 
 {\displaystyle M}
 
 is called complemented if it has a topological complement 
 
 
 
 N
 
 
 {\displaystyle N}
 
 (and uncomplemented if not). The choice of 
 
 
 
 N
 
 
 {\displaystyle N}
 
 can matter quite strongly: every complemented vector subspace 
 
 
 
 M
 
 
 {\displaystyle M}
 
 has algebraic complements that do not complement 
 
 
 
 M
 
 
 {\displaystyle M}
 
 topologically. 
Because a <a href="/facts/Linear_map/Q1PBKNVV">linear map</a> between two <a href="/facts/Normed_space/rmDljfyE">normed</a> (or <a href="/facts/Banach_space/sDEIk7EC">Banach</a>) spaces is <a href="/facts/Bounded_linear_map/LYmDTIqR">bounded</a> if and only if it is <a href="/facts/Continuous_linear_map/KcqjDBrJ">continuous</a>, the definition in the categories of normed (resp. <a href="/facts/Banach_space/sDEIk7EC">Banach</a>) spaces is the same as in topological vector spaces.

<h3>Equivalent characterizations</h3>
The vector subspace 
 
 
 
 M
 
 
 {\displaystyle M}
 
 is complemented in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 if and only if any of the following holds:<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>

<ul><li>There exists a continuous linear map 
 
 
 
 
 P
 
 M
 
 
 :
 X
 →
 X
 
 
 {\displaystyle P_{M}:X\to X}
 
 with <a href="/facts/Image_(mathematics)/KANefMAl">image</a> 
 
 
 
 
 P
 
 M
 
 
 (
 X
 )
 =
 M
 
 
 {\displaystyle P_{M}(X)=M}
 
 such that 
 
 
 
 P
 ∘
 P
 =
 P
 
 
 {\displaystyle P\circ P=P}
 
. That is, 
 
 
 
 
 P
 
 M
 
 
 
 
 {\displaystyle P_{M}}
 
 is a <a href="/facts/Continuous_function/sKbl02pB">continuous</a> <a href="/facts/Linear_map/Q1PBKNVV">linear</a> <a href="/facts/Projection_(mathematics)/QxxC9XTp">projection</a> onto 
 
 
 
 M
 
 
 {\displaystyle M}
 
. (In that case, algebraically 
 
 
 
 X
 =
 M
 ⊕
 ker
 ⁡
 
 P
 
 
 
 {\displaystyle X=M\oplus \ker {P}}
 
, and it is the continuity of 
 
 
 
 
 P
 
 M
 
 
 
 
 {\displaystyle P_{M}}
 
 that implies that this is a complement.)</li>
<li>For every TVS 
 
 
 
 Y
 ,
 
 
 {\displaystyle Y,}
 
 the <a href="/facts/Restriction_map/lwVzsKIt">restriction map</a> 
 
 
 
 R
 :
 L
 (
 X
 ;
 Y
 )
 →
 L
 (
 M
 ;
 Y
 )
 ;
 R
 (
 u
 )
 =
 u
 
 
 |
 
 
 M
 
 
 
 
 {\displaystyle R:L(X;Y)\to L(M;Y);R(u)=u|_{M}}
 
 is surjective.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a></li></ul>
If in addition 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is <a href="/facts/Banach_space/sDEIk7EC">Banach</a>, then an equivalent condition is

<ul><li>
 
 
 
 M
 
 
 {\displaystyle M}
 
 is <a href="/facts/Closure_(topology)/9x2wlgUW">closed</a> in 
 
 
 
 X
 
 
 {\displaystyle X}
 
, there exists another closed subspace 
 
 
 
 N
 ⊆
 X
 
 
 {\displaystyle N\subseteq X}
 
, and 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is an <a href="/facts/Banach_space/sDEIk7EC">isomorphism</a> from the <a href="/facts/Direct_sum_of_Banach_spaces/Y6PBJnjg">abstract direct sum</a> 
 
 
 
 M
 ⊕
 N
 
 
 {\displaystyle M\oplus N}
 
 to 
 
 
 
 X
 
 
 {\displaystyle X}
 
.</li></ul>
<h2 id="examples">Examples</h2>
<ul><li>If 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 is a measure space and 
 
 
 
 X
 ⊆
 Y
 
 
 {\displaystyle X\subseteq Y}
 
 has positive measure, then 
 
 
 
 
 L
 
 p
 
 
 (
 X
 )
 
 
 {\displaystyle L^{p}(X)}
 
 is complemented in 
 
 
 
 
 L
 
 p
 
 
 (
 Y
 )
 
 
 {\displaystyle L^{p}(Y)}
 
.</li>
<li>
 
 
 
 
 c
 
 0
 
 
 
 
 {\displaystyle c_{0}}
 
, the space of sequences converging to 
 
 
 
 0
 
 
 {\displaystyle 0}
 
, is complemented in 
 
 
 
 c
 
 
 {\displaystyle c}
 
, the space of convergent sequences.</li>
<li>By <a href="/facts/Lebesgue_decomposition/E8lHD2BF">Lebesgue decomposition</a>, 
 
 
 
 
 L
 
 1
 
 
 (
 [
 0
 ,
 1
 ]
 )
 
 
 {\displaystyle L^{1}([0,1])}
 
 is complemented in 
 
 
 
 
 r
 c
 a
 
 (
 [
 0
 ,
 1
 ]
 )
 ≅
 C
 (
 [
 0
 ,
 1
 ]
 
 )
 
 ∗
 
 
 
 
 {\displaystyle \mathrm {rca} ([0,1])\cong C([0,1])^{*}}
 
.</li></ul>
<h2 id="sufficient-conditions">Sufficient conditions</h2>
For any two topological vector spaces 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
, the subspaces 
 
 
 
 X
 ×
 {
 0
 }
 
 
 {\displaystyle X\times \{0\}}
 
 and 
 
 
 
 {
 0
 }
 ×
 Y
 
 
 {\displaystyle \{0\}\times Y}
 
 are topological complements in 
 
 
 
 X
 ×
 Y
 
 
 {\displaystyle X\times Y}
 
. 
Every algebraic complement of 
 
 
 
 
 
 
 {
 0
 }
 
 ¯
 
 
 
 
 {\displaystyle {\overline {\{0\}}}}
 
, the closure of 
 
 
 
 0
 
 
 {\displaystyle 0}
 
, is also a topological complement. This is because 
 
 
 
 
 
 
 {
 0
 }
 
 ¯
 
 
 
 
 {\displaystyle {\overline {\{0\}}}}
 
 has the <a href="/facts/Indiscrete_topology/uA7RlGth">indiscrete topology</a>, and so the algebraic projection is continuous.<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a>
If 
 
 
 
 X
 =
 M
 ⊕
 N
 
 
 {\displaystyle X=M\oplus N}
 
 and 
 
 
 
 A
 :
 X
 →
 Y
 
 
 {\displaystyle A:X\to Y}
 
 is surjective, then 
 
 
 
 Y
 =
 A
 M
 ⊕
 A
 N
 
 
 {\displaystyle Y=AM\oplus AN}
 
.<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a>

<h3>Finite dimension</h3>
Suppose 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is Hausdorff and <a href="/facts/Locally_convex_topological_vector_space/AGrPrayC">locally convex</a> and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 a <a href="/facts/Free_vector_space/o6rC6yoJ">free topological vector subspace</a>: for some set 
 
 
 
 I
 
 
 {\displaystyle I}
 
, we have 
 
 
 
 Y
 ≅
 
 
 K
 
 
 I
 
 
 
 
 {\displaystyle Y\cong \mathbb {K} ^{I}}
 
 (as a t.v.s.). Then 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 is a closed and complemented vector subspace of 
 
 
 
 X
 
 
 {\displaystyle X}
 
.<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a> In particular, any finite-dimensional subspace of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is complemented.<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a> 
In arbitrary topological vector spaces, a finite-dimensional vector subspace 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 is topologically complemented if and only if for every non-zero 
 
 
 
 y
 ∈
 Y
 
 
 {\displaystyle y\in Y}
 
, there exists a continuous linear functional on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 that <a href="/facts/Separating_set/I8VjYAgQ">separates</a> 
 
 
 
 y
 
 
 {\displaystyle y}
 
 from 
 
 
 
 0
 
 
 {\displaystyle 0}
 
.<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a> For an example in which this fails, see § Fréchet spaces.

<h3>Finite codimension</h3>
Not all finite-<a href="/facts/Codimension/AlGab3eJ">codimensional</a> vector subspaces of a TVS are closed, but those that are, do have complements.<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a><a class="footnote-ref" id="fnref:19" href="#fn:19">19</a>

<h3>Hilbert spaces</h3>
In a <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a>, the <a href="/facts/Orthogonal_complement/a13oQBsz">orthogonal complement</a> 
 
 
 
 
 M
 
 ⊥
 
 
 
 
 {\displaystyle M^{\bot }}
 
 of any closed vector subspace 
 
 
 
 M
 
 
 {\displaystyle M}
 
 is always a topological complement of 
 
 
 
 M
 
 
 {\displaystyle M}
 
. This property characterizes Hilbert spaces within the class of <a href="/facts/Banach_space/sDEIk7EC">Banach spaces</a>: every infinite dimensional, non-Hilbert Banach space contains a closed uncomplemented subspace, a deep theorem of <a href="/facts/Joram_Lindenstrauss/LW2BeVTd">Joram Lindenstrauss</a> and Lior Tzafriri.<a class="footnote-ref" id="fnref:20" href="#fn:20">20</a><a class="footnote-ref" id="fnref:21" href="#fn:21">21</a>

<h3>Fréchet spaces</h3>
Let 
 
 
 
 X
 
 
 {\displaystyle X}
 
 be a <a href="/facts/Fr%25C3%25A9chet_space/9gXkQu2u">Fréchet space</a> over the field 
 
 
 
 
 K
 
 
 
 {\displaystyle \mathbb {K} }
 
. Then the following are equivalent:<a class="footnote-ref" id="fnref:22" href="#fn:22">22</a>

<ol><li>
 
 
 
 X
 
 
 {\displaystyle X}
 
 is not normable (that is, any continuous norm does not generate the topology)</li>
<li>
 
 
 
 X
 
 
 {\displaystyle X}
 
 contains a vector subspace TVS-isomorphic to 
 
 
 
 
 
 K
 
 
 
 N
 
 
 
 .
 
 
 {\displaystyle \mathbb {K} ^{\mathbb {N} }.}
 
</li>
<li>
 
 
 
 X
 
 
 {\displaystyle X}
 
 contains a complemented vector subspace TVS-isomorphic to 
 
 
 
 
 
 K
 
 
 
 N
 
 
 
 
 
 {\displaystyle \mathbb {K} ^{\mathbb {N} }}
 
.</li></ol>
<h2 id="properties-examples-of-uncomplemented-subspaces">Properties; examples of uncomplemented subspaces</h2>
A complemented (vector) subspace of a Hausdorff space 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is necessarily a <a href="/facts/Closed_set/upYnRTUj">closed subset</a> of 
 
 
 
 X
 
 
 {\displaystyle X}
 
, as is its complement.<a class="footnote-ref" id="fnref:23" href="#fn:23">23</a><a class="footnote-ref" id="fnref:24" href="#fn:24">24</a>
From the existence of <a href="/facts/Hamel_bases/89IPoN6c">Hamel bases</a>, every infinite-dimensional Banach space contains unclosed linear subspaces.<a class="footnote-ref" id="fnref:25" href="#fn:25">25</a> Since any complemented subspace is closed, none of those subspaces is complemented. 
Likewise, if 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a <a href="/facts/Complete_topological_vector_space/e6qtS76j">complete TVS</a> and 
 
 
 
 X
 
 /
 
 M
 
 
 {\displaystyle X/M}
 
 is not complete, then 
 
 
 
 M
 
 
 {\displaystyle M}
 
 has no topological complement in 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 
<a class="footnote-ref" id="fnref:26" href="#fn:26">26</a>

<h2 id="applications">Applications</h2>
If 
 
 
 
 A
 :
 X
 →
 Y
 
 
 {\displaystyle A:X\to Y}
 
 is a continuous linear <a href="/facts/Surjection/KezhLpCb">surjection</a>, then the following conditions are equivalent:

<ol><li>The kernel of 
 
 
 
 A
 
 
 {\displaystyle A}
 
 has a topological complement.</li>
<li>There exists a "right inverse": a continuous linear map 
 
 
 
 B
 :
 Y
 →
 X
 
 
 {\displaystyle B:Y\to X}
 
 such that 
 
 
 
 A
 B
 =
 
 
 I
 d
 
 
 Y
 
 
 
 
 {\displaystyle AB=\mathrm {Id} _{Y}}
 
, where 
 
 
 
 
 Id
 
 Y
 
 
 :
 Y
 →
 Y
 
 
 {\displaystyle \operatorname {Id} _{Y}:Y\to Y}
 
 is the identity map.<a class="footnote-ref" id="fnref:27" href="#fn:27">27</a></li></ol>
(Note: This claim is an erroneous exercise given by Trèves. Let 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 both be 
 
 
 
 
 R
 
 
 
 {\displaystyle \mathbb {R} }
 
 where 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is endowed with the usual topology, but 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 is endowed with the trivial topology. The identity map 
 
 
 
 X
 →
 Y
 
 
 {\displaystyle X\to Y}
 
 is then a continuous, linear bijection but its inverse is not continuous, since 
 
 
 
 X
 
 
 {\displaystyle X}
 
 has a finer topology than 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
. The kernel 
 
 
 
 {
 0
 }
 
 
 {\displaystyle \{0\}}
 
 has 
 
 
 
 X
 
 
 {\displaystyle X}
 
 as a topological complement, but we have just shown that no continuous right inverse can exist. If 
 
 
 
 A
 :
 X
 →
 Y
 
 
 {\displaystyle A:X\to Y}
 
 is also open (and thus a TVS homomorphism) then the claimed result holds.)

<h3>The Method of Decomposition</h3>
Topological vector spaces admit the following <a href="/facts/Cantor-Schroder-Bernstein_theorem/uKUbvFpo">Cantor-Schröder-Bernstein–type theorem</a>:

Let 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 be TVSs such that 
 
 
 
 X
 =
 X
 ⊕
 X
 
 
 {\displaystyle X=X\oplus X}
 
 and 
 
 
 
 Y
 =
 Y
 ⊕
 Y
 .
 
 
 {\displaystyle Y=Y\oplus Y.}
 
 Suppose that 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 contains a complemented copy of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 X
 
 
 {\displaystyle X}
 
 contains a complemented copy of 
 
 
 
 Y
 .
 
 
 {\displaystyle Y.}
 
 Then 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is TVS-isomorphic to 
 
 
 
 Y
 .
 
 
 {\displaystyle Y.}

The "self-splitting" assumptions that 
 
 
 
 X
 =
 X
 ⊕
 X
 
 
 {\displaystyle X=X\oplus X}
 
 and 
 
 
 
 Y
 =
 Y
 ⊕
 Y
 
 
 {\displaystyle Y=Y\oplus Y}
 
 cannot be removed: <a href="/facts/Tim_Gowers/El3xoEmE">Tim Gowers</a> showed in 1996 that there exist non-isomorphic <a href="/facts/Banach_space/sDEIk7EC">Banach spaces</a> 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
, each complemented in the other.<a class="footnote-ref" id="fnref:28" href="#fn:28">28</a>

<h2 id="in-classical-banach-spaces">In classical Banach spaces</h2>
Understanding the complemented subspaces of an arbitrary Banach space 
 
 
 
 X
 
 
 {\displaystyle X}
 
 <a href="/facts/Up_to/ydz4ztzX">up to</a> isomorphism is a classical problem that has motivated much work in basis theory, particularly the development of absolutely summing operators. The problem remains open for a variety of important Banach spaces, most notably the space 
 
 
 
 
 L
 
 1
 
 
 [
 0
 ,
 1
 ]
 
 
 {\displaystyle L_{1}[0,1]}
 
.<a class="footnote-ref" id="fnref:29" href="#fn:29">29</a> 
For some Banach spaces the question is closed. Most famously, if 
 
 
 
 1
 ≤
 p
 ≤
 ∞
 
 
 {\displaystyle 1\leq p\leq \infty }
 
 then the only complemented infinite-dimensional subspaces of 
 
 
 
 
 ℓ
 
 p
 
 
 
 
 {\displaystyle \ell _{p}}
 
 are isomorphic to 
 
 
 
 
 ℓ
 
 p
 
 
 ,
 
 
 {\displaystyle \ell _{p},}
 
 and the same goes for 
 
 
 
 
 c
 
 0
 
 
 .
 
 
 {\displaystyle c_{0}.}
 
 Such spaces are called prime (when their only infinite-dimensional complemented subspaces are isomorphic to the original). These are not the only prime spaces, however.<a class="footnote-ref" id="fnref:30" href="#fn:30">30</a>
The spaces 
 
 
 
 
 L
 
 p
 
 
 [
 0
 ,
 1
 ]
 
 
 {\displaystyle L_{p}[0,1]}
 
 are not prime whenever 
 
 
 
 p
 ∈
 (
 1
 ,
 2
 )
 ∪
 (
 2
 ,
 ∞
 )
 ;
 
 
 {\displaystyle p\in (1,2)\cup (2,\infty );}
 
 in fact, they admit uncountably many non-isomorphic complemented subspaces.<a class="footnote-ref" id="fnref:31" href="#fn:31">31</a> 
The spaces 
 
 
 
 
 L
 
 2
 
 
 [
 0
 ,
 1
 ]
 
 
 {\displaystyle L_{2}[0,1]}
 
 and 
 
 
 
 
 L
 
 ∞
 
 
 [
 0
 ,
 1
 ]
 
 
 {\displaystyle L_{\infty }[0,1]}
 
 are isomorphic to 
 
 
 
 
 ℓ
 
 2
 
 
 
 
 {\displaystyle \ell _{2}}
 
 and 
 
 
 
 
 ℓ
 
 ∞
 
 
 ,
 
 
 {\displaystyle \ell _{\infty },}
 
 respectively, so they are indeed prime.<a class="footnote-ref" id="fnref:32" href="#fn:32">32</a>
The space 
 
 
 
 
 L
 
 1
 
 
 [
 0
 ,
 1
 ]
 
 
 {\displaystyle L_{1}[0,1]}
 
 is not prime, because it contains a complemented copy of 
 
 
 
 
 ℓ
 
 1
 
 
 
 
 {\displaystyle \ell _{1}}
 
. No other complemented subspaces of 
 
 
 
 
 L
 
 1
 
 
 [
 0
 ,
 1
 ]
 
 
 {\displaystyle L_{1}[0,1]}
 
 are currently known.<a class="footnote-ref" id="fnref:33" href="#fn:33">33</a>

<h2 id="indecomposable-banach-spaces">Indecomposable Banach spaces</h2>
An infinite-dimensional Banach space is called indecomposable whenever its only complemented subspaces are either finite-dimensional or -codimensional. Because a finite-<a href="/facts/Codimension/AlGab3eJ">codimensional</a> subspace of a Banach space 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is always isomorphic to 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 indecomposable Banach spaces are prime.
The most well-known example of indecomposable spaces are in fact hereditarily indecomposable, which means every infinite-dimensional subspace is also indecomposable.<a class="footnote-ref" id="fnref:34" href="#fn:34">34</a> 

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Direct_sum/x3ljVsP6">Direct sum</a> – Operation in abstract algebra composing objects into "more complicated" objects</li>
<li><a href="/facts/Direct_sum_of_modules/Y6PBJnjg">Direct sum of modules</a> – Operation in abstract algebra</li>
<li><a href="/facts/Direct_sum_of_topological_groups/rezTtyTf">Direct sum of topological groups</a></li></ul>
<h2 id="proofs">Proofs</h2>

<h2 id="bibliography">Bibliography</h2>
<ul><li>Bachman, George; Narici, Lawrence (2000). Functional Analysis (Second ed.). Mineola, New York: Dover Publications. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0486402512. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/829157984">829157984</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>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>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/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><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="references">References</h2>

<ol>
<li id="fn:1">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:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Fabian, Marián J.; Habala, Petr; Hájek, Petr; Montesinos Santalucía, Vicente; Zizler, Václav (2011). Banach Space Theory: The Basis for Linear and Nonlinear Analysis (PDF). New York: Springer. pp. 179–181. doi:10.1007/978-1-4419-7515-7. ISBN 978-1-4419-7515-7. <a href="978-1-4419-7515-7" target="_blank">978-1-4419-7515-7</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Fabian, Marián J.; Habala, Petr; Hájek, Petr; Montesinos Santalucía, Vicente; Zizler, Václav (2011). Banach Space Theory: The Basis for Linear and Nonlinear Analysis (PDF). New York: Springer. pp. 179–181. doi:10.1007/978-1-4419-7515-7. ISBN 978-1-4419-7515-7. <a href="978-1-4419-7515-7" target="_blank">978-1-4419-7515-7</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Brezis, Haim (2011). Functional Analysis, Sobolev Spaces, and Partial Differential Equations. Universitext. New York: Springer. pp. 38–39. ISBN 978-0-387-70913-0. <a href="978-0-387-70913-0" target="_blank">978-0-387-70913-0</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Schaefer & Wolff 1999, pp. 19–24. - 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:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Fabian, Marián J.; Habala, Petr; Hájek, Petr; Montesinos Santalucía, Vicente; Zizler, Václav (2011). Banach Space Theory: The Basis for Linear and Nonlinear Analysis (PDF). New York: Springer. pp. 179–181. doi:10.1007/978-1-4419-7515-7. ISBN 978-1-4419-7515-7. <a href="978-1-4419-7515-7" target="_blank">978-1-4419-7515-7</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">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:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">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:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Fabian, Marián J.; Habala, Petr; Hájek, Petr; Montesinos Santalucía, Vicente; Zizler, Václav (2011). Banach Space Theory: The Basis for Linear and Nonlinear Analysis (PDF). New York: Springer. pp. 179–181. doi:10.1007/978-1-4419-7515-7. ISBN 978-1-4419-7515-7. <a href="978-1-4419-7515-7" target="_blank">978-1-4419-7515-7</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">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:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">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:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Trèves 2006, p. 36. - 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:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">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:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Fabian, Marián J.; Habala, Petr; Hájek, Petr; Montesinos Santalucía, Vicente; Zizler, Václav (2011). Banach Space Theory: The Basis for Linear and Nonlinear Analysis (PDF). New York: Springer. pp. 179–181. doi:10.1007/978-1-4419-7515-7. ISBN 978-1-4419-7515-7. <a href="978-1-4419-7515-7" target="_blank">978-1-4419-7515-7</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Y
 
 
 {\displaystyle Y}
 
 is closed because 
 
 
 
 
 
 K
 
 
 I
 
 
 
 
 {\displaystyle \mathbb {K} ^{I}}
 
 is complete and 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is Hausdorff.

Let 
 
 
 
 f
 =
 
 
 (
 
 f
 
 i
 
 
 )
 
 
 i
 ∈
 I
 
 
 :
 Y
 →
 
 
 K
 
 
 I
 
 
 
 
 {\displaystyle f=\left(f_{i}\right)_{i\in I}:Y\to \mathbb {K} ^{I}}
 
 be a TVS-isomorphism; each 
 
 
 
 
 f
 
 i
 
 
 :
 Y
 →
 
 K
 
 
 
 {\displaystyle f_{i}:Y\to \mathbb {K} }
 
 is a continuous linear functional. By the Hahn–Banach theorem, we may extend each 
 
 
 
 
 f
 
 i
 
 
 
 
 {\displaystyle f_{i}}
 
 to a continuous linear functional 
 
 
 
 
 F
 
 i
 
 
 :
 X
 →
 
 K
 
 
 
 {\displaystyle F_{i}:X\to \mathbb {K} }
 
 on 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 
 The joint map 
 
 
 
 F
 :
 X
 →
 
 
 K
 
 
 I
 
 
 
 
 {\displaystyle F:X\to \mathbb {K} ^{I}}
 
 is a continuous linear surjection whose restriction to 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 is 
 
 
 
 f
 
 
 {\displaystyle f}
 
. The composition 
 
 
 
 P
 =
 
 f
 
 −
 1
 
 
 ∘
 F
 :
 X
 →
 Y
 
 
 {\displaystyle P=f^{-1}\circ F:X\to Y}
 
 is then a continuous continuous projection onto 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
.Q.E.D. <a href="/wiki/Complete_(topology)" target="_blank">/wiki/Complete_(topology)</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Rudin 1991, p. 106. - 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:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">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:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">Rudin 1991, p. 106. - 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:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Serre, Jean-Pierre (1955). "Un théoreme de dualité". Commentarii Mathematici Helvetici. 29 (1): 9–26. doi:10.1007/BF02564268. S2CID 123643759. <a href="/wiki/Commentarii_Mathematici_Helvetici" target="_blank">/wiki/Commentarii_Mathematici_Helvetici</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">Lindenstrauss, J., & Tzafriri, L. (1971). On the complemented subspaces problem. Israel Journal of Mathematics, 9, 263-269. <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
<li id="fn:21">Brezis, Haim (2011). Functional Analysis, Sobolev Spaces, and Partial Differential Equations. Universitext. New York: Springer. pp. 38–39. ISBN 978-0-387-70913-0. <a href="978-0-387-70913-0" target="_blank">978-0-387-70913-0</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></li>
<li id="fn:22">Jarchow 1981, pp. 129–130. - 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:22" class="footnote-back-ref">↩</a></li>
<li id="fn:23">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:23" class="footnote-back-ref">↩</a></li>
<li id="fn:24">In a Hausdorff space, 
 
 
 
 {
 0
 }
 
 
 {\displaystyle \{0\}}
 
 is closed. A complemented space is the kernel of the (continuous) projection onto its complement. Thus it is the preimage of 
 
 
 
 {
 0
 }
 
 
 {\displaystyle \{0\}}
 
 under a continuous map, and so closed.Q.E.D. <a href="/wiki/Q.E.D." target="_blank">/wiki/Q.E.D.</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></li>
<li id="fn:25">Any sequence 
 
 
 
 {
 
 e
 
 j
 
 
 
 }
 
 j
 =
 0
 
 
 ∞
 
 
 ∈
 
 X
 
 ω
 
 
 
 
 {\displaystyle \{e_{j}\}_{j=0}^{\infty }\in X^{\omega }}
 
 defines a summation map 
 
 
 
 T
 :
 
 l
 
 1
 
 
 →
 X
 ;
 T
 (
 {
 
 x
 
 j
 
 
 
 }
 
 j
 
 
 )
 =
 
 ∑
 
 j
 
 
 
 
 x
 
 j
 
 
 
 e
 
 j
 
 
 
 
 
 {\displaystyle T:l^{1}\to X;T(\{x_{j}\}_{j})=\sum _{j}{x_{j}e_{j}}}
 
. But if 
 
 
 
 {
 
 e
 
 j
 
 
 
 }
 
 j
 
 
 
 
 {\displaystyle \{e_{j}\}_{j}}
 
 are (algebraically) linearly independent and 
 
 
 
 {
 
 x
 
 j
 
 
 
 }
 
 j
 
 
 
 
 {\displaystyle \{x_{j}\}_{j}}
 
 has full support, then 
 
 
 
 T
 (
 x
 )
 ∈
 
 
 
 span
 ⁡
 
 {
 
 e
 
 j
 
 
 
 }
 
 j
 
 
 
 
 ¯
 
 
 ∖
 span
 ⁡
 
 {
 
 e
 
 j
 
 
 
 }
 
 j
 
 
 
 
 
 {\displaystyle T(x)\in {\overline {\operatorname {span} {\{e_{j}\}_{j}}}}\setminus \operatorname {span} {\{e_{j}\}_{j}}}
 
. Q.E.D. <a href="/wiki/Q.E.D." target="_blank">/wiki/Q.E.D.</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></li>
<li id="fn:26">Schaefer & Wolff 1999, pp. 190–202. - 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:26" class="footnote-back-ref">↩</a></li>
<li id="fn:27">Trèves 2006, p. 36. - 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:27" class="footnote-back-ref">↩</a></li>
<li id="fn:28">Narici & Beckenstein 2011, pp. 100–101. - 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>
<li id="fn:29">Albiac, Fernando; Kalton, Nigel J. (2006). Topics in Banach Space Theory. GTM 233 (2nd ed.). Switzerland: Springer (published 2016). pp. 29–232. doi:10.1007/978-3-319-31557-7. ISBN 978-3-319-31557-7. <a href="978-3-319-31557-7" target="_blank">978-3-319-31557-7</a> <a href="#fnref:29" class="footnote-back-ref">↩</a></li>
<li id="fn:30">Albiac, Fernando; Kalton, Nigel J. (2006). Topics in Banach Space Theory. GTM 233 (2nd ed.). Switzerland: Springer (published 2016). pp. 29–232. doi:10.1007/978-3-319-31557-7. ISBN 978-3-319-31557-7. <a href="978-3-319-31557-7" target="_blank">978-3-319-31557-7</a> <a href="#fnref:30" class="footnote-back-ref">↩</a></li>
<li id="fn:31">Albiac, Fernando; Kalton, Nigel J. (2006). Topics in Banach Space Theory. GTM 233 (2nd ed.). Switzerland: Springer (published 2016). pp. 29–232. doi:10.1007/978-3-319-31557-7. ISBN 978-3-319-31557-7. <a href="978-3-319-31557-7" target="_blank">978-3-319-31557-7</a> <a href="#fnref:31" class="footnote-back-ref">↩</a></li>
<li id="fn:32">Albiac, Fernando; Kalton, Nigel J. (2006). Topics in Banach Space Theory. GTM 233 (2nd ed.). Switzerland: Springer (published 2016). pp. 29–232. doi:10.1007/978-3-319-31557-7. ISBN 978-3-319-31557-7. <a href="978-3-319-31557-7" target="_blank">978-3-319-31557-7</a> <a href="#fnref:32" class="footnote-back-ref">↩</a></li>
<li id="fn:33">Albiac, Fernando; Kalton, Nigel J. (2006). Topics in Banach Space Theory. GTM 233 (2nd ed.). Switzerland: Springer (published 2016). pp. 29–232. doi:10.1007/978-3-319-31557-7. ISBN 978-3-319-31557-7. <a href="978-3-319-31557-7" target="_blank">978-3-319-31557-7</a> <a href="#fnref:33" class="footnote-back-ref">↩</a></li>
<li id="fn:34">Argyros, Spiros; Tolias, Andreas (2004). Methods in the Theory of Hereditarily Indecomposable Banach Spaces. American Mathematical Soc. ISBN 978-0-8218-3521-0. <a href="978-0-8218-3521-0" target="_blank">978-0-8218-3521-0</a> <a href="#fnref:34" class="footnote-back-ref">↩</a></li>
</ol>

Complemented subspace open-in-new

Complemented subspace