Bessel inequality

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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 $x$ ).

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Theorem

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