Definition

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

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.

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 $M^{\perp}$ , i.e.
  
• 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.