Characterization of Bases (Hilbert Spaces)

Theorem
Let $H$ be a Hilbert space, and let $E$ be an orthonormal subset of $H$.

Then the following four statements are equivalent:


 * $(1): \qquad E$ is a basis for $H$
 * $(2): \qquad h \in H, h \perp E \implies h = \mathbf 0$, where $\perp$ denotes orthogonality
 * $(3): \qquad \vee E = H$, where $\vee E$ denotes the closed linear span of $E$
 * $(4): \qquad \forall h \in H: h = \sum \left\{{\left\langle{h, e}\right\rangle}e: e \in E\right\}$, where $\Sigma$ denotes a generalized sum