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:

$(1): \qquad E$ 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 $E$

$(4): \qquad \forall h \in H: h = \displaystyle \sum \left\{{\left\langle{h, e}\right\rangle e: e \in E}\right\}$

$(5): \qquad \forall g, h \in H: \left\langle{g, h}\right\rangle = \displaystyle \sum \left\{{\left\langle{g, e}\right\rangle\left\langle{e, h}\right\rangle: e \in E}\right\}$

$(6): \qquad \forall h \in H: \left\Vert{h}\right\Vert^2 = \displaystyle \sum \left\{{\left\vert{\left\langle{h, e}\right\rangle}\right\vert^2: e \in E}\right\}$

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

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

Sources

• 1990: John B. Conway: A Course in Functional Analysis ... (previous) ... (next) $I.4.12-13$