Characterization of Bases (Hilbert Spaces)

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

Then the following six statements are equivalent:


 * $(1): \quad E$ is a basis for $H$
 * $(2): \quad h \in H, h \perp E \implies h = \mathbf 0$, where $\perp$ denotes orthogonality
 * $(3): \quad \vee E = H$, where $\vee E$ denotes the closed linear span of $E$
 * $(4): \quad \forall h \in H: h = \ds \sum \set {\innerprod h e e: e \in E}$
 * $(5): \quad \forall g, h \in H: \innerprod g h = \ds \sum \set {\innerprod g e \innerprod e h: e \in E}$
 * $(6): \quad \forall h \in H: \norm h^2 = \ds \sum \set {\size {\innerprod h e}^2: e \in E}$

In the last three statements, $\ds \sum$ denotes a generalized sum.

Statement $(6)$ is commonly known as Parseval's identity.