Let $H$ be a Hilbert space and $\left\{ e_{i}\right\} _{i=1}^{\infty}$ an orthonormal system. I need to prove that the following set is a convex set:
$C=\left\{ x\in H\,:\,\sum_{n=1}^{\infty}\left(1+\frac{1}{n}\right)^{2}\cdot\left|\left\langle x,e_{n}\right\rangle \right|^{2}\leq1\right\}.$
I thought to define: $C_{k}=\left\{ x\in H\,:\,\sum_{n=1}^{k}\left(1+\frac{1}{n}\right)^{2}\cdot\left|\left\langle x,e_{n}\right\rangle \right|^{2}\leq1\right\}$ and to prove that $C_{k}$ is convex for all $k$ and then we have $C=\bigcap_k C_{k}$ which will be convex too, but i couldn't manage to prove it. I would be glad to get some help.
Thanks!