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.
Best Answer
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.