Complemented subspace

Remarks

Equivalently, $M$ is complemented if and only if there is a continuous linear map $P \colon X \to X$ with $P^2 = P$ and $P(X) = M$, a bounded projection onto $M$. Given such $P$, take $N = \ker P$; given $N$, the algebraic projection onto $M$ along $N$ has closed graph, so it is continuous by the closed graph theorem. Example: in a Hilbert space every closed subspace $M$ is complemented by its orthogonal complement $M^\perp$, which is closed with $M \cap M^\perp = \set{0}$ and $M + M^\perp$ equal to the whole space. Every finite-dimensional subspace of a Banach space is complemented: extend a dual basis by the Hahn–Banach theorem to build the projection. Non-example: $c_0$ is a closed subspace of $\ell^\infty$ that is not complemented; every closed subspace $N$ with $c_0 \cap N = \set{0}$ has $c_0 + N \ne \ell^\infty$. A theorem of Lindenstrauss and Tzafriri says that a Banach space in which every closed subspace is complemented is isomorphic to a Hilbert space.