# Bessel's Inequality

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## Theorem

Let $H$ be a Hilbert space.

Let $E = \set {e_n: n \in \N}$ be a countably infinite orthonormal subset of $H$.

Then, for all $h \in H$:

- $\ds \sum_{n \mathop = 1}^\infty \size {\innerprod h {e_n} }^2 \le \norm h^2$

### Corollary 1

If $E$ is an orthonormal subset of $H$, then for any $h \in H$, the set $\ds \set {e_n \in E: \innerprod h {e_n} \ne 0}$ is countable.

### Corollary 2

The condition in the theorem that the orthonormal set $E$ be countable is superfluous.

## Proof

## Source of Name

This entry was named for Friedrich Wilhelm Bessel.

## Sources

- 1990: John B. Conway:
*A Course in Functional Analysis*... (previous) ... (next): $\text I.4.8-10$