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:

$(1): \quad E$ 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.

Proof

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Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next) $\text {I.4.12-13}$