Bessel's Inequality

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 $h \in H$:


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

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

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