# 贝塞尔不等式

## 定理的叙述

${\displaystyle {\mathcal {H}}}$ 是一个装备了内积${\displaystyle \left\langle \cdot ,\cdot \right\rangle }$希尔伯特空间。考虑一组规范正交向量序列${\displaystyle (e_{1},e_{2},\cdots ,e_{n},\cdots )}$。那么，对于任意一个${\displaystyle {\mathcal {H}}}$ 中的元素，都有：

${\displaystyle \sum _{k}\left|\left\langle x,e_{k}\right\rangle \right|^{2}\leq \left\|x\right\|^{2}}$

## 证明

${\displaystyle (e_{1},e_{2},\cdots ,e_{n})}$

${\displaystyle \langle p(x),z(x)\rangle =\langle \sum _{k=1}^{n}\langle x,e_{k}\rangle e_{k},x-\sum _{k=1}^{n}\langle x,e_{k}\rangle e_{k}\rangle =\sum _{k=1}^{n}(\langle x,e_{k}\rangle \langle e_{k},x\rangle )-\sum _{k=1}^{n}\sum _{l=1}^{n}\langle x,e_{k}\rangle {\overline {\langle x,e_{l}\rangle }}\langle e_{k},e_{l}\rangle }$
${\displaystyle =\sum _{k=1}^{n}|\langle x,e_{k}\rangle |^{2}-\sum _{k=1}^{n}|\langle x,e_{k}\rangle |^{2}\langle e_{k},e_{k}\rangle =\sum _{k=1}^{n}|\langle x,e_{k}\rangle |^{2}-\sum _{k=1}^{n}|\langle x,e_{k}\rangle |^{2}=0}$

${\displaystyle \left\|x\right\|^{2}=\left\|p(x)\right\|^{2}+\left\|z(x)\right\|^{2}\geq \left\|p(x)\right\|^{2}=\sum _{k=1}^{n}|\langle x,e_{k}\rangle |^{2}\langle e_{k},e_{k}\rangle =\sum _{k=1}^{n}|\langle x,e_{k}\rangle |^{2}}$