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

• 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 $X$ is an Hilbert space, then each closed subspace $M\subseteq X$ is topologically complemented by its orthogonal complement $M^{⟂}$, i.e.

  $$M\oplus M^{⟂}=X.$$

• 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 Lindenstrauss-Tzafriri theorem (http://planetmath.org/CharacterizationOfAHilbertSpace).