 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.

In functional analysis and related fields of mathematics, a **complemented subspace** is a type of subspace of a vector space that has a certain structure with respect to the entire space. More specifically, consider a vector space \( V \) and a subspace \( W \subseteq V \).