Bessel inequality

Let $\mathscr{H}$ be a Hilbert space, and suppose $e_1,e_2,\mathrm{\dots }\in \mathscr{H}$ is an orthonormal sequence. Then for any $x\in \mathscr{H}$,

$$\sum _{k=1}^{\mathrm{\infty }}{\left|⟨x,e_k⟩\right|}^2\le {\parallel x\parallel }^2.$$

Bessel’s inequality immediately lets us define the sum

$$x^{\prime }=\sum _{k=1}^{\mathrm{\infty }}⟨x,e_k⟩e_k.$$

The inequality means that the series converges.

For a complete orthonormal series, we have Parseval’s theorem, which replaces inequality with equality (and consequently $x^{\prime }$ with $x$).