\begin{definition}[Definition:Basis (Hilbert Space)]
Let $H$ be a Hilbert space.
A '''basis for $H$''' is a maximal orthonormal subset of $H$.
Thus, $B$ is a '''basis''' for $H$ {{iff}} for all orthonormal subsets $B'$ of $H$:
:$B \subseteq B' \implies B = B'$
\end{definition}