I want to show that for $(a_n)$ a bounded series in $\mathbb{R}$, $x_n := \frac{1}{n} \sum_{k= 0}^n a_k e_k$ converges in norm (induced by the inner product) to 0, where $e_n$ is an orthonormal basis of the Hilbert space.
If $c$ is the upper bound for $(a_n)$, I can show that
$||x_n||^2 = \frac{1}{n} \big|\big|\sum_{k = 0}^n a_k e_k\big|\big| = \frac{1}{n} \sum_{j,k=0}^n a_j a_k^* \langle e_j,e_k\rangle = \frac{1}{n} \sum_{k= 0}^n |a_k|^2 \leq c^2 \frac{1}{n}(n+1)$.
But this converges to $c^2$ and not to 0. So now I am not sure how to find a stronger estimate in order to show my claim.
Best Answer
Actually, for real $a_n$,
$$\|x_n\|^2=\frac{1}{n^2}\sum_{k=0}^na_k^2\leq \frac{c^2}{n}$$