Show that if $(〈x_n,y〉)_{n \in \mathbb{N}}$ converges for all $y \in H$, then there is $x \in H$ such that $f(x_n) \to f(x)$ as $n \to \infty$

Question

Consider a sequence $(x_n)_{n \in \mathbb{N}}$ in a Hilbert space $H$. Show that if $(〈x_n,y〉)_{n \in \mathbb{N}}$ converges for all $y \in H$, then there is $x \in H$ such that $f(x_n) \to f(x)$ as $n \to \infty$, for all $f \in H'$, where $H'$ is the dual.

Good day for everyone, I don't know how I can solve it. Can you help me?

$\lim_{n\to \infty} || <x_n,y>|| = \lim_{n\to\infty} (||\sum x_n\cdot y||)^{1/2} = y $

Answer 1

Let $\varphi:y\rightarrow\lim_{n}\left<y,x_{n}\right>$, Uniform Boundedness Principle gives $\varphi\in H'$, and hence by Riesz Representation Theorem for some $x$, $\varphi(y)=\left<y,x\right>$.

Now given $f\in H'$, by Riesz Representation Theorem again for some $y$, $f(z)=\left<z,y\right>$, then $\lim_{n}f(x_{n})=\lim_{n}\left<x_{n},y\right>=\overline{\varphi(y)}=\overline{\left<y,x\right>}=\left<x,y\right>=f(x)$.