Bessel's Inequality

Theorem
Let $H$ be a Hilbert space, and let $E = \left\{{e_n: n \in \N}\right\}$ be a countably infinite orthonormal subset of $H$.

Then, for all $h \in H$, one has the inequality:


 * $\displaystyle \sum_{n \mathop = 1}^\infty \left|{\left\langle{h, e_n}\right\rangle}\right|^2 \le \left\|{h}\right\|^2$

Corollary 1
If $E$ is an orthonormal subset of $H$, then for any $h \in H$, the set $\displaystyle \left\{{e_n \in E: \left\langle{h, e_n}\right\rangle \ne 0}\right\}$ is countable.

Corollary 2
The condition in the theorem that the orthonormal set $E$ be countable is superfluous.