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 \( W \subseteq V \).
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A closed vector subspace of a Banach space is complemented if a bounded linear operator satisfies and . Equivalently, is the topological direct sum of and a closed subspace: a projection gives , and the coordinate projection of such a decomposition is bounded.