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Let $V$ be an Euclidean space and $(e_1,e_2,\dots,e_n)$ an orthonormal vector system of $V$. Show that, for every $x \in V$ the following is valid:

$$\sum_{i=1}^n (e_i|x)^2 \leq \| x\|^2. $$

Can someone help me?

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    What's $e_i|x$? Do you mean $\langle e_i , x\rangle$? – 2017-01-16
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    It may be the same thing, notations... I mean the inner product – 2017-01-16

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$(e_1,e_2,\ldots,e_n)$ can be extended to an orthonormal bases $(e_1,e_2,\ldots,e_m)$. Let $a_1,a_2,\ldots,a_m$ be real numbers such that $x=a_1e_1+\ldots+a_me_m$. Clearly, $a_i=(e_i|x)$. Then, $$\|x\|^2=\|e_1a_1+\ldots+e_ma_m\|^2=\sum_{i=1}^m\|e_i\|^2a_i^2+\sum_{i\ne j}(e_i|e_j)a_ia_j=\sum_{i=1}^m a_i^2\ge \sum_{i=1}^n a_i^2$$ The last equation comes from the fact that $(e_1,e_2,\ldots,e_m)$ is an orthonormal basis.