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In the branch of [[mathematics]] called [[functional analysis]], a <!-- The literature rarely (if ever) prefixes "(un)complemented subspace" with the qualifier "topologically."  Every vector subspace <math>M</math> of <math>V</math> has an algebraic complement: another vector subspace <math>N</math> such that <math>V=M\oplus N</math> as an algebraic direct sum.  But such constructions are not only trivial; they are useless in functional analysis.  Consequently the term is unambiguous.   --> '''complemented subspace''' of a [[topological vector space]] <math>X,</math> is a [[vector subspace]] <math>M</math> for which there exists some other vector subspace <math>N</math> of <math>X,</math> called its ('''topological''') '''complement''' in <math>X</math>, such that <math>X</math> is the [[Direct sum of topological vector spaces|direct sum <math>M \oplus N</math> in the category of topological vector spaces]].   Formally, topological direct sums strengthen the [[Direct sum of modules|algebraic direct sum]] by requiring certain maps be continuous; the result retains many nice properties from the operation of direct sum in finite-dimensional vector spaces.   

Every finite-dimensional subspace of a [[Banach space]] is complemented, but other subspaces may not.  In general, classifying all complemented subspaces is a difficult problem, which has been solved only for [[List of Banach spaces|some well-known Banach spaces]].     

The concept of a complemented subspace is analogous to, but distinct from, that of a [[Complement (set theory)|set complement]].  The set-theoretic complement of a vector subspace is never a complementary subspace.

==Preliminaries: definitions and notation==

If <math>X</math> is a vector space and <math>M</math> and <math>N</math> are [[vector subspace]]s of <math>X</math> then there is a well-defined addition map
<math display="block">\begin{alignat}{4}
S :\;&& M \times N &&\;\to    \;& X \\
     && (m, n)     &&\;\mapsto\;& m + n \\
\end{alignat}</math>
The map <math>S</math> is a [[morphism]] in the [[category of vector spaces]]&nbsp;— that is to say, [[linear map|linear]]. 

===Algebraic direct sum===
{{Main|Direct sum|Direct sum of modules}}

The vector space <math>X</math> is said to be the algebraic [[direct sum]] (or direct sum in the category of vector spaces) <math>M\oplus N</math> when any of the following equivalent conditions are satisfied:
#The addition map <math>S : M \times N \to X</math> is a [[vector space isomorphism]].{{sfn|Grothendieck|1973|pp=34-36}}<ref name="FabianRevised">{{cite book |author=Fabian |first=Marián&nbsp;J. |url=https://link.springer.com/content/pdf/10.1007/978-1-4419-7515-7.pdf |title=Banach Space Theory: The Basis for Linear and Nonlinear Analysis |last2=Habala |first2=Petr |last3=Hájek |first3=Petr |last4=Montesinos&nbsp;Santalucía |first4=Vicente |last5=Zizler |first5=Václav |date=2011 |publisher=Springer |isbn=978-1-4419-7515-7 |location=New York |pages=179–181|doi=10.1007/978-1-4419-7515-7 }}</ref>
#The addition map is bijective.
#<math>M \cap N = \{0\}</math> and <math>M + N = X</math>; in this case <math>N</math> is called an '''algebraic complement''' or '''supplement''' to <math>M</math> in <math>X</math> and the two subspaces are said to be '''complementary''' or '''supplementary'''.<ref name="FabianRevised" /><ref name=":0">{{cite book |last=Brezis |first=Haim |title=Functional Analysis, Sobolev Spaces, and Partial Differential Equations |publisher=Springer |year=2011 |isbn=978-0-387-70913-0 |series=Universitext |location=New York |pages=38–39}}</ref>

When these conditions hold, the inverse <math>S^{-1} : X \to M \times N</math> is well-defined and can be written in terms of coordinates as<math display="block">S^{-1} = \left(P_M, P_N\right)\text{.}</math>
The first coordinate <math>P_M : X \to M</math> is called the canonical projection of <math>X</math> onto <math>M</math>; likewise the second coordinate is the canonical projection onto <math>N.</math>{{sfn|Schaefer|Wolff|1999|pp=19-24}} 

Equivalently, <math>P_M(x)</math> and <math>P_N(x)</math> are the unique vectors in <math>M</math> and <math>N,</math> respectively, that satisfy
<math display="block">x = P_M(x) + P_N(x)\text{.}</math>
As maps, <math display="block">P_M + P_N = \operatorname{Id}_X, \qquad \ker P_M = N, \qquad \text{ and } \qquad \ker P_N = M</math> where <math>\operatorname{Id}_X</math> denotes the [[identity map]] on <math>X</math>.<ref name="FabianRevised" />

== Motivation ==
{{See also|Coproduct|Direct sum#Direct sum in categories|Direct sum of topological groups}}

Suppose that the vector space <math>X</math> is the algebraic direct sum of <math>M\oplus N</math>.  In the category of vector spaces, finite [[Product (category theory)|products]] and [[Coproduct|coproducts]] coincide: algebraically, <math>M \oplus N</math> and <math>M \times N</math> are indistinguishable.  Given a problem involving elements of <math>X</math>, one can break the elements down into their components in <math>M</math> and <math>N</math>, because the projection maps defined above act as inverses to the natural inclusion of <math>M</math> and <math>N</math> into <math>X</math>.  Then one can solve the problem in the vector subspaces and recombine to form an element of <math>X</math>.  

In the category of [[topological vector space|topological vector spaces]], that algebraic decomposition becomes less useful.   The definition of a topological vector space requires the addition map <math>S</math> to be continuous; its inverse <math>S^{-1} : X \to M \times N</math> may not be.{{sfn|Grothendieck|1973|pp=34-36}}  The [[Coproduct|categorical definition of direct sum]], however, requires <math>P_M</math> and <math>P_N</math> to be morphisms&nbsp;— that is, ''continuous'' linear maps.  

The space <math>X</math> is the '''topological direct sum''' of <math>M</math> and <math>N</math> if (and only if) any of the following equivalent conditions hold:
#The addition map <math>S : M \times N \to X</math> is a [[TVS-isomorphism]] (that is, a surjective [[Linear map|linear]] [[homeomorphism]]).{{sfn|Grothendieck|1973|pp=34-36}}
#<math>X</math> is the algebraic direct sum of <math>M</math> and <math>N</math> and also any of the following equivalent conditions: {{ordered list | list-style-type = lower-latin
|The inverse of the addition map <math>S^{-1} : X \to M \times N</math> is continuous.
|Both canonical projections <math>P_M : X \to M</math> and <math>P_N : X \to N</math> are continuous.
|At least one of the canonical projections <math>P_M</math> and <math>P_N</math> is continuous.
|{{anchor|ComplementIsQuotient}}The canonical [[quotient map]] <math>p : N \to X / M; p(n)=n+M</math> is an isomorphism of topological vector spaces (i.e. a linear homeomorphism).<ref name=FabianRevised />}}
#<math>X</math> is the [[Coproduct|direct sum]] of <math>M</math> and <math>N</math> in the category of topological vector spaces.
#The map <math>S</math> is [[bijective]] and [[open map|open]].
#When considered as [[Additive group|additive]] [[topological group]]s, <math>X</math> is the [[Direct sum of topological groups|topological direct sum of the subgroups]] <math>M</math> and <math>N.</math>
The topological direct sum is also written <math>X = M \oplus N</math>; whether the sum is in the topological or algebraic sense is usually clarified through [[Context (language use)|context]].

== Definition ==
Every topological direct sum is an algebraic direct sum <math>X = M \oplus N</math>; the converse is not guaranteed.  Even if both <math>M</math> and <math>N</math> are closed in <math>X</math>, <math>S^{-1}</math> may ''still'' fail to be continuous.  <math>N</math> is a (topological) '''complement''' or '''supplement''' to <math>M</math> if it avoids that pathology&nbsp;— that is, if, topologically, <math>X = M \oplus N</math>.  (Then <math>M</math>&nbsp;is likewise complementary to <math>N</math>.){{sfn|Grothendieck|1973|pp=34-36}}  [[#ComplementIsQuotient|Condition 2(d) above]] implies that any topological complement of <math>M</math> is isomorphic, as a topological vector space, to the [[quotient vector space]] <math>X / M</math>.

<math>M</math> is called '''complemented''' if it has a topological complement <math>N</math> (and '''uncomplemented''' if not).  The choice of <math>N</math> can matter quite strongly: every complemented vector subspace <math>M</math> has algebraic complements that do not complement <math>M</math> topologically.   

Because a [[linear map]] between two [[Normed space|normed]] (or [[Banach space|Banach]]) spaces is [[Bounded linear map|bounded]] if and only if it is [[Continuous linear map|continuous]], the definition in the categories of normed (resp. [[Banach space|Banach]]) spaces is the same as in topological vector spaces.

=== Equivalent characterizations ===
The vector subspace <math>M</math> is complemented in <math>X</math> if and only if any of the following holds:{{sfn|Grothendieck|1973|pp=34-36}}
*There exists a continuous linear map <math>P_M : X \to X</math> with [[Image (mathematics)|image]] <math>P_M(X) = M</math> such that <math>P \circ P = P</math>.  That is, <math>P_M</math> is a {{Em|[[Continuous function|continuous]]}} [[linear map|linear]] [[Projection (mathematics)|projection]] onto <math>M</math>.  (In that case, {{Em|algebraically}} <math>X=M\oplus\ker{P}</math>, and it is the continuity of <math>P_M</math> that implies that this is a complement.)
* For every TVS <math>Y,</math> the [[restriction map]] <math>R : L(X; Y) \to L(M; Y); R(u)=u|_M</math> is surjective.{{sfn|Trèves|2006|p=36}}
If in addition <math>X</math> is [[Banach space|Banach]], then an equivalent condition is

* <math>M</math> is [[Closure (topology)|closed]] in <math>X</math>, there exists another closed subspace <math>N\subseteq X</math>, and <math>S</math> is an [[Banach space#Linear operators, isomorphisms|isomorphism]] from the [[Direct sum of Banach spaces|abstract direct sum]] <math>M \oplus N</math> to <math>X</math>.

==Examples==
* If <math>Y</math> is a measure space and <math>X\subseteq Y</math> has positive measure, then <math>L^p(X)</math> is complemented in <math>L^p(Y)</math>.  
* <math>c_0</math>, the space of sequences converging to <math>0</math>, is complemented in <math>c</math>, the space of convergent sequences. 
* By [[Lebesgue decomposition]], <math>L^1([0,1])</math> is complemented in <math>\mathrm{rca}([0,1])\cong C([0,1])^*</math>.  

==Sufficient conditions==
For any two topological vector spaces <math>X</math> and <math>Y</math>, the subspaces <math>X \times \{0\}</math> and <math>\{0\} \times Y</math> are topological complements in <math>X \times Y</math>.   <!-- If neither X nor Y are Hausdorff then neither X×{0} nor {0}×Y is a closed subspace of X×Y, but that is not required by the definition.  -->

Every algebraic complement of <math>\overline{\{0\}}</math>, the closure of <math>0</math>, is also a topological complement.  This is because <math>\overline{\{0\}}</math> has the [[indiscrete topology]], and so the algebraic projection is continuous.{{sfn|Wilansky|2013|p=63}}

If <math>X=M\oplus N</math> and <math>A:X\to Y</math> is surjective, then <math>Y=AM\oplus AN</math>.<ref name="FabianRevised" />

=== Finite dimension ===
Suppose <math>X</math> is Hausdorff and [[Locally convex topological vector space|locally convex]] and <math>Y</math> a [[Free vector space|free topological vector subspace]]: for some set <math>I</math>, we have <math>Y\cong\mathbb{K}^I</math> (as a t.v.s.).  Then <math>Y</math> is a closed and complemented vector subspace of <math>X</math>.<ref group="proof"><math>Y</math> is closed because <math>\mathbb{K}^I</math> is [[Complete (topology)|complete]] and <math>X</math> is Hausdorff.

<div>Let <math>f = \left(f_i\right)_{i \in I} : Y \to \mathbb{K}^I</math> be a TVS-isomorphism; each <math>f_i : Y \to \mathbb{K}</math> is a continuous linear functional. By the [[Hahn–Banach theorem]], we may extend each <math>f_i</math> to a continuous linear functional <math>F_i : X \to \mathbb{K}</math> on <math>X.</math>  The [[joint map]] <math>F : X \to \mathbb{K}^I</math> is a continuous linear surjection whose restriction to <math>Y</math> is <math>f</math>.  The composition <math>P = f^{-1} \circ F : X \to Y</math> is then a continuous continuous projection onto <math>Y</math>.{{align|right|[[Q.E.D.]]}}</div></ref>  In particular, any finite-dimensional subspace of <math>X</math> is complemented.{{sfn|Rudin|1991|p=106}}    

In arbitrary topological vector spaces, a finite-dimensional vector subspace <math>Y</math> is topologically complemented if and only if for every non-zero <math>y\in Y</math>, there exists a continuous linear functional on <math>X</math> that [[Separating set|separates]] <math>y</math> from <math>0</math>.{{sfn|Grothendieck|1973|pp=34-36}}  For an example in which this fails, see {{Slink||Fréchet spaces}}.

=== Finite codimension ===
Not all finite-[[codimension]]al vector subspaces of a TVS are closed, but those that are, do have complements.{{sfn|Rudin|1991|p=106}}<ref>{{cite journal |last1=Serre |first1=Jean-Pierre |date=1955 |title=Un théoreme de dualité |journal=[[Commentarii Mathematici Helvetici]] |volume=29|issue=1 |pages=9–26 |doi=10.1007/BF02564268 |s2cid=123643759 }}</ref>

=== Hilbert spaces ===
In a [[Hilbert space]], the [[orthogonal complement]] <math>M^{\bot}</math> of any closed vector subspace <math>M</math> is always a topological complement of <math>M</math>.  This property characterizes Hilbert spaces within the class of [[Banach space|Banach spaces]]: every infinite dimensional, non-Hilbert Banach space contains a closed uncomplemented subspace, a deep theorem of [[Joram Lindenstrauss]] and [[Lior Tzafriri]].<ref>Lindenstrauss, J., & Tzafriri, L. (1971). On the complemented subspaces problem. Israel Journal of Mathematics, 9, 263-269.</ref><ref name=":0" />

=== Fréchet spaces ===
Let <math>X</math> be a [[Fréchet space]] over the field <math>\mathbb{K}</math>.  Then the following are equivalent:{{sfn|Jarchow|1981|pp=129-130}}

# <math>X</math> is not normable (that is, any continuous norm does not generate the topology)
# <math>X</math> contains a vector subspace TVS-isomorphic to <math>\mathbb{K}^{\N}.</math>
# <math>X</math> contains a complemented vector subspace TVS-isomorphic to <math>\mathbb{K}^{\N}</math>.

==Properties; examples of uncomplemented subspaces==
A complemented (vector) subspace of a Hausdorff space <math>X</math> is necessarily a [[Closed set|closed subset]] of <math>X</math>, as is its complement.{{sfn|Grothendieck|1973|pp=34-36}}<ref name="CompSubspaceOfHausdorffSTVSisClosed" group="proof" /><!-- In non-Hausdorff topological vector spaces, closure is not usually added to the definition. -->

From the existence of [[Hamel bases]], every infinite-dimensional Banach space contains unclosed linear subspaces.<ref group=proof>Any sequence <math>\{e_j\}_{j=0}^{\infty}\in X^{\omega}</math> defines a summation map <math>T:l^1\to X; T(\{x_j\}_j)=\sum_j{x_je_j}</math>.  But if <math>\{e_j\}_j</math> are (algebraically) linearly independent and <math>\{x_j\}_j</math> has full support, then <math>T(x)\in\overline{\operatorname{span}{\{e_j\}_j}}\setminus\operatorname{span}{\{e_j\}_j}</math>.  {{align|right|[[Q.E.D.]]}}</ref>  Since any complemented subspace is closed, none of those subspaces is complemented.  

Likewise, if <math>X</math> is a [[Complete topological vector space|complete TVS]] and <math>X / M</math> is not complete, then <math>M</math> has no topological complement in <math>X.</math>{{sfn|Schaefer|Wolff|1999|pp=190-202}}

== Applications ==
If <math>A : X \to Y</math> is a continuous linear [[surjection]], then the following conditions are equivalent:
# The kernel of <math>A</math> has a topological complement.
# There exists a "right inverse": a continuous linear map <math>B : Y \to X</math> such that <math>AB = \mathrm{Id}_Y</math>, where <math>\operatorname{Id}_Y : Y \to Y</math> is the identity map.{{sfn|Trèves|2006|p=36}}
(Note: This claim is an erroneous exercise given by Trèves. Let <math>X</math> and <math>Y</math> both be <math>\mathbb{R}</math> where <math>X</math> is endowed with the usual topology, but <math>Y</math> is endowed with the trivial topology. The identity map <math>X \to Y</math> is then a continuous, linear bijection but its inverse is not continuous, since <math>X</math> has a finer topology than <math>Y</math>. The kernel <math>\{0\}</math> has <math>X</math> as a topological complement, but we have just shown that no continuous right inverse can exist.  If <math>A: X \to Y</math> is also open (and thus a TVS homomorphism) then the claimed result holds.)

=== The Method of Decomposition ===
Topological vector spaces admit the following [[Cantor-Schroder-Bernstein theorem|Cantor-Schröder-Bernstein–type theorem]]:
:Let <math>X</math> and <math>Y</math> be TVSs such that <math>X = X \oplus X</math> and <math>Y = Y \oplus Y.</math> Suppose that <math>Y</math> contains a complemented copy of <math>X</math> and <math>X</math> contains a complemented copy of <math>Y.</math> Then <math>X</math> is TVS-isomorphic to <math>Y.</math>
The "self-splitting" assumptions that <math>X = X \oplus X</math> and <math>Y = Y \oplus Y</math> cannot be removed: [[Tim Gowers]] showed in 1996 that there exist non-isomorphic [[Banach space]]s <math>X</math> and <math>Y</math>, each complemented in the other.{{sfn|Narici|Beckenstein|2011|pp=100-101}}

==In classical Banach spaces==

Understanding the complemented subspaces of an arbitrary Banach space <math>X</math> [[up to]] isomorphism is a classical problem that has motivated much work in basis theory, particularly the development of [[Absolutely summing operator|absolutely summing operators]].  The problem remains open for a variety of important Banach spaces, most notably the space <math>L_1[0,1]</math>.<ref name=":1">{{Cite book |last1=Albiac |first1=Fernando |url=https://link.springer.com/book/10.1007/978-3-319-31557-7 |title=Topics in Banach Space Theory |last2=Kalton |first2=Nigel&nbsp;J. |publisher=Springer |year=2006 |isbn=978-3-319-31557-7 |edition=2nd |series=GTM 233 |location=Switzerland |publication-date=2016 |pages=29–232 |doi=10.1007/978-3-319-31557-7}}</ref>  

For some Banach spaces the question is closed. Most famously, if <math>1 \leq p \leq \infty</math> then the only complemented infinite-dimensional subspaces of <math>\ell_p</math> are isomorphic to <math>\ell_p,</math> and the same goes for <math>c_0.</math>  Such spaces are called {{em|prime}} (when their only infinite-dimensional complemented subspaces are isomorphic to the original). These are not the only prime spaces, however.<ref name=":1" />

The spaces <math>L_p[0,1]</math> are not prime whenever <math>p \in (1, 2) \cup (2, \infty);</math> in fact, they admit uncountably many non-isomorphic complemented subspaces.<ref name=":1" /> 

The spaces <math>L_2[0,1]</math> and <math>L_{\infty}[0,1]</math> are isomorphic to <math>\ell_2</math> and <math>\ell_{\infty},</math> respectively, so they are indeed prime.<ref name=":1" />

The space <math>L_1[0,1]</math> is not prime, because it contains a complemented copy of <math>\ell_1</math>.  No other complemented subspaces of <math>L_1[0,1]</math> are currently known.<ref name=":1" />

==Indecomposable Banach spaces==

An infinite-dimensional Banach space is called '''indecomposable''' whenever its only complemented subspaces are either finite-dimensional or -codimensional.  Because a finite-[[codimension]]al subspace of a Banach space <math>X</math> is always isomorphic to <math>X,</math> indecomposable Banach spaces are prime.

The most well-known example of indecomposable spaces are in fact {{em|hereditarily}} indecomposable, which means every infinite-dimensional subspace is also indecomposable.<ref>{{Cite book |last1=Argyros |first1=Spiros |url=https://books.google.com/books?id=I93UCQAAQBAJ&dq=indecomposable+banach+space&pg=PR5 |title=Methods in the Theory of Hereditarily Indecomposable Banach Spaces |last2=Tolias |first2=Andreas |date=2004 |publisher=American Mathematical Soc. |isbn=978-0-8218-3521-0 |language=en}}</ref> 

==See also==

* {{annotated link|Direct sum}}
* {{annotated link|Direct sum of modules}}
* {{annotated link|Direct sum of topological groups}}

== Proofs ==
{{reflist|group=proof|refs=
<ref group=proof name=CompSubspaceOfHausdorffSTVSisClosed>In a Hausdorff space, <math>\{0\}</math> is closed.  A complemented space is the kernel of the (continuous) projection onto its complement.  Thus it is the preimage of <math>\{0\}</math> under a continuous map, and so closed.{{align|right|[[Q.E.D.]]}}
</ref>
}}

==References==

{{reflist}}

==Bibliography==

* {{cite book
| last1=Bachman
| first1=George
| last2=Narici
| first2=Lawrence
| title=Functional Analysis
| edition=Second
| publisher=Dover Publications
| location=Mineola, New York
| year=2000
| series=
| isbn=978-0486402512
| oclc=829157984
}} <!--{{sfn|Bachman|Narici|2000|p=}}-->
* {{Grothendieck Topological Vector Spaces}} <!--{{sfn|Grothendieck|1973|p=}}-->
* {{Jarchow Locally Convex Spaces}} <!--{{sfn|Jarchow|1981|p=}}-->
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!--{{sfn|Narici|Beckenstein|2011|p=}}-->
* {{Rudin Walter Functional Analysis|edition=2}} <!--{{sfn|Rudin|1991|p=}}-->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!--{{sfn|Schaefer|Wolff|1999|p=}}-->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!--{{sfn|Trèves|2006|p=}}-->
* {{Wilansky Modern Methods in Topological Vector Spaces|edition=1}}

{{TopologicalVectorSpaces}}
{{Functional analysis}}

[[Category:Functional analysis]]