Show that the sequence $x_N$ converges weakly and compute the weak limit Let $H$ be a Hilbert space and let $\{ e_k \}_{k = 1}^\infty $ be an orthonormal basis for $H$. I am trying to prove that the sequence
$$x_N = \frac{1}{\sqrt N}\sum_{k = 1}^N e_k $$
converges weakly and to find its weak limit.
What I know so far:
Proving weak convergence of $u_N$ to $u$ means
proving that $\ell (u_N) \to \ell(u)$ for all functionals $\ell$ defined as $\ell :H \to \mathbb R$. In a Hilbert space, however, linear functionals take the form $\ell(u_N) = \langle u,u_N \rangle $ for some unique $u$ that changes only when $\ell$ changes.
Could someone give me any clues in proving the above, i.e., that the weak limit exist and how to find it.
 A: Observe that $\left\langle x_N, e_j \right\rangle = \frac{1}{\sqrt{N}} \to 0$, as $N \to \infty$. Thus, by linearity, for any $y = a_1 e_1 + \ldots + a_r e_r$, we have $\left\langle x_N, e_j \right\rangle \to 0$. Let $y \in H$, with $y = \sum_{n=1}^\infty a_n e_n$ and $\epsilon > 0$. Get $R$ so large that $\| y - \sum_{k=1}^R  a_n e_n\|_H < \epsilon$. Then,
$$|\left\langle x_N, y \right\rangle| \leq \left| \left\langle x_N, \sum_{k=1}^R  a_n e_n \right\rangle  \right| + \left| \left\langle x_N, \sum_{k=R+1}^\infty  a_n e_n \right\rangle \right| \leq \underbrace{\left| \left\langle x_N, \sum_{k=1}^R  a_n e_n \right\rangle  \right|}_{\to 0 } + \epsilon$$
Where the last inequality is Cauchy-Schwartz with $\|x_N\| = 1$. We may send $N \to \infty$ and conclude to $x_N$ converges to zero weakly.
A: $\{x_n\}$ converges weakly to 0.
It suffices to show that
$$
\langle x_n,y\rangle\to 0,
$$
for all $y\in H$.
Let $y\in H$, then $y=\sum_{n=1}^\infty a_ne_n$, where
$a_n=\langle y,e_n\rangle$, and $\|y\|^2=\sum_{n=1}^\infty a_n^2$.
Hence
$$
\langle x_n,y\rangle=\left\langle \frac{1}{n}\sum_{k=1}^ne_k,\sum_{n=1}^\infty a_ne_n\right\rangle=\frac{1}{n}\sum_{k=1}^n a_k.
$$
Clearly, since the sequence $\{a^n\}$ is summable, then $a^2\to 0$, and hence $a_n\to 0$ as well. So does the sequence
$$
b_n=\frac{1}{n}\sum_{k=1}^n a_k.
$$
See for example Stolz–Cesàro theorem
