Characterization of Bases (Hilbert Spaces)

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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.


Proof


Sources