# Bessel inequality

Let $\mathcal{H}$ be a Hilbert space, and suppose $e_{1},e_{2},\ldots\in\mathcal{H}$ is an orthonormal sequence. Then for any $x\in\mathcal{H}$,

 $\sum_{k=1}^{\infty}\left|\left\langle x,e_{k}\right\rangle\right|^{2}\leq\left% \|x\right\|^{2}.$

Bessel’s inequality immediately lets us define the sum

 $x^{\prime}=\sum_{k=1}^{\infty}\left\langle x,e_{k}\right\rangle e_{k}.$

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 $x$).

Title Bessel inequality BesselInequality 2013-03-22 12:46:38 2013-03-22 12:46:38 ariels (338) ariels (338) 5 ariels (338) Theorem msc 46C05