= Closed subspace of a Hilbert space
{title2=$\overline V=V$}
= Closed subspaces of a Hilbert space
{synonym}
A <vector subspace> closed in the <norm topology> of a <Hilbert space> is complete in the inherited <norm> and <inner product>. Its <orthogonal complement> gives a decomposition of the ambient <Hilbert space> into an <orthogonal direct sum>. This closedness permits <orthogonal projection> and ensures limits used in range arguments remain in the subspace. Every finite-dimensional <vector subspace> is closed.
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