topological complement
0.0.1 Definition
Let $X$ be a topological vector space^{} and $M\subseteq X$ a closed (http://planetmath.org/ClosedSet) subspace^{}.
If there exists a closed subspace $N\subseteq X$ such that
$$M\oplus N=X$$ 
we say that $M$ is topologically complemented.
In this case $N$ is said to be a topological complement of $M$, and also $M$ and $N$ are said to be topologically complementary subspaces.
0.0.2 Remarks

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It is known that every subspace $M\subseteq X$ has an algebraic complement, i.e. there exists a subspace $N\subseteq X$ such that $M\oplus N=X$. The existence of topological complements, however, is not always assured.

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If $X$ is an Hilbert space^{}, then each closed subspace $M\subseteq X$ is topologically complemented by its orthogonal complement^{} ${M}^{\u27c2}$, i.e.
$$M\oplus {M}^{\u27c2}=X.$$ 
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Moreover, for Banach spaces^{} the converse^{} of the last paragraph also holds, i.e. if each closed subspace is topologically complemented then $X$ is isomorphic a Hilbert space. This is the LindenstraussTzafriri theorem (http://planetmath.org/CharacterizationOfAHilbertSpace).
Title  topological complement 

Canonical name  TopologicalComplement 
Date of creation  20130322 17:32:31 
Last modified on  20130322 17:32:31 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  5 
Author  asteroid (17536) 
Entry type  Definition 
Classification  msc 46A99 
Classification  msc 15A03 
Defines  topologically complementary 
Defines  topologically complemented 