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topological complement

(Definition)

Let

If there exists a closed subspace $N \subseteq X$ such that

$\displaystyle M \oplus N = X$

we say that

is topologically complemented.

In this case is said to be a topological complement of , and also and are said to be topologically complementary subspaces.

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.
If is an Hilbert space, then each closed subspace $M \subseteq X$ is topologically complemented by its orthogonal complement $M^{\perp}$ , i.e.
$\displaystyle M \oplus M^{\perp} = X .$
Moreover, for Banach spaces the converse of the last paragraph also holds, i.e. if each closed subspace is topologically complemented then is a Hilbert space. This is the Lindenstrauss-Tzafriri theorem.

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Also defines:

topologically complementary, topologically complemented

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AMS MSC:	15A03 (Linear and multilinear algebra; matrix theory :: Vector spaces, linear dependence, rank)
	46A99 (Functional analysis :: Topological linear spaces and related structures :: Miscellaneous)

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