Definition:Basis (Hilbert Space)

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Let $H$ be a Hilbert space.

A basis for $H$ is a maximal orthonormal subset of $H$.

Thus, $B$ is a basis for $H$ if and only if for all orthonormal subsets $B'$ of $H$:

$B \subseteq B' \implies B = B'$

Linguistic Note

The plural of basis is bases.

This is properly pronounced bay-seez, not bay-siz.