# 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) $\text {I.4.12-13}$