An application of Riesz representation theorem $\lim_{n\rightarrow \infty} \left< x_n,e_k\right>=\left< x,e_k \right>$.

Question

Suppose that $\mathcal{H}$ be a real, separable Hilbert space and $\{e_k \}$ be a countable orthonormal basis. And let $x\in \mathcal{H}$ and $\{x_n\}\subset \mathcal{H}$ is a bounded sequence. Than prove that $$\lim_{n\rightarrow \infty}\left< x_n,e_k\right>=\left<x,e_k\right>\hspace{3mm}\forall k \implies \lim_{n\rightarrow \infty} \psi(x_n)=\psi(x ) \forall \psi\in \mathcal{H}^* $$

I am trying to prove the statement above. I was trying to work on only finite sum of orthonormal basis but I don;t know how to connect to infinite sum after that. I hope to be able to get some hint or answer. Thank you in advance :)

Answer 1

Without loss of generality, suppose $x=0$. We first show that $\lim_{n \rightarrow \infty}\langle x_n,y \rangle=0$ for all $y \in H$. Let $y=\sum_{i=1}^\infty t_ie_i$. Given $\epsilon>0$, choose $m$ large enough so that $\|y-y_m\|<\epsilon$, where $y_m=\sum_{i=1}^m t_ie_i$. Then $$|\langle x_n,y_m \rangle|\leq \sum_{i=1}^m |t_i| |\langle x_n,e_i\rangle |,$$ and so for $n$ large enough $|\langle x_n,y_m \rangle|<\epsilon$. Now for $n$ large enough $$|\langle x_n,y \rangle-\langle x_n, y_m \rangle| =|\langle x_n, y-y_m \rangle|<\|x_n\| \|y-y_m\| \leq C \epsilon,$$ where $C$ is the bound on the sequence $x_i$. Therefore, for $n$ large enough $$|\langle x_n, y \rangle|< |\langle x_n,y_m \rangle|+C\epsilon<(C+1)\epsilon.$$ Since $\epsilon$ was arbitrary, we have $\lim \langle x_n,y \rangle=0$ for all $y$. The rest follows from Riesz Representation Theorem.

Answer 2

For any $y \in H$ we have $|\sum_N^{\infty} \langle x_n,e_k \rangle \langle y,e_k \rangle| \leq (\sum_N^{\infty} (\langle x_n,e_k)^{2})^{1/2} (\sum_N^{\infty} (\langle y,e_k)^{2})^{1/2}$. The first factor is bounded by $\|x_n\|$ which is bounded in $n$. Hence, given $\epsilon>0$ we can choose $N$ such that $|\sum_N^{\infty} \langle x_n,e_k \rangle \langle y,e_k \rangle| <\epsilon $ for all $n$. We can choose $N$ such that we also have $|\sum_N^{\infty} \langle y,e_k \rangle \langle y,e_k \rangle| <\epsilon $. Can you now complete the proof by splitting sum over all $k$ into sum from $1$ to $N-1$ and the one from $N$ to $\infty$?