Closed linear subspace

A linear subspace $M$ of a normed vector space $E$ is a closed linear subspace if it is a closed subset of $E$ in the norm topology.

Equivalent characterization and properties

Equivalently, if a sequence of points of $M$ converges to $x\in E$ , then $x\in M$ .

A closed linear subspace of a Banach space is again a Banach space. In a Hilbert space, every closed subspace $M$ has an orthogonal complement and every vector decomposes uniquely as $x=m+m^\perp$ .