Problem II.3.13: How to prove $\sum_{k=0}^{m}|(x | u_{k})|^{2} \le \|x\|^{2}$ for an orthonormal system of an inner product space

Question

I'm doing Problem II.3.13.a in textbook Analysis I by Amann/Escher.

in which

$B=\left\{u_{0}, \ldots, u_{m}\right\}$ is an orthonormal system in an inner product space $(E,(\cdot | \cdot ))$ and $F :=\operatorname{span}(B)$.

I tried to use the CS inequality as hinted, but I get a weaker inequality:

$$\sum_{k=0}^{m} |(x | u_{k})|^{2} \le \sum_{k=0}^{m} \|x\|^2 \|u_k\|^2 = \sum_{k=0}^{m} \|x\|^2 = (m+1)\|x\|^2$$

Please help me solve (a). Thank you so much!

Answer 1

After some manipulation, we have \begin{align*} \sum_{k=0}^m|(x\mid u_k)|^2&=(x\mid\sum_{k=0}^m(x\mid u_k)u_k)\\ &\leq\|x\|\cdot\left\|\sum_{k=0}^m(x\mid u_k)u_k\right\|\\ &=\|x\|\left(\sum_{k=0}^m|(x\mid u_k)|^2\right)^{1/2}. \end{align*} Rearranging, we obtain the result.