Complemented subspace
= Complemented subspace
A closed <vector subspace> $Y$ of a <Banach space> $X$ is complemented if a <bounded linear operator> $P:X\to X$ satisfies $P^2=P$ and $P(X)=Y$. Equivalently, $X$ is the topological <direct sum> of $Y$ and a closed subspace: a projection gives $X=Y\oplus\ker P$, and the coordinate projection of such a decomposition is bounded.