Hilbert space and weak convergence

Question

I have some dubts about this exercise:

Let $(H, (\cdot, \cdot))$ a Hilbert space e let $(x_{n})_{n}\subset H$ and $x \in H$. Prove that

1. If $x_{n}\xrightarrow{\tau_{w}}x$ if and only if $(x_{n},y) \rightarrow (x,y)$ for any $y\in H$.
2. If $x_{n}\rightarrow x$ if and only if $x_{n}\xrightarrow{\tau_{w}}x$ and $\| x_{n} \| \rightarrow \|x \|$.

$\tau_{w}$ is the weak convergence, hence $x_{n}\xrightarrow{\tau_{w}}x$ if and only if $(f,x_{n}) \rightarrow (f,x_{n})$ for any $f\in H'$.

For the second point I argued as following. If $x_{n}\rightarrow x$ then I know \begin{equation} \lim_{n\rightarrow +\infty} \|x_{n}-x\| \rightarrow 0, \end{equation} hence I have that $\| x_{n} \| \rightarrow \|x \|$. Now for the weak convergence I considered \begin{equation} |(f,x_{n}-x)| \leq \|f\|_{H'} \| x_{n}-x\|_{H} \rightarrow 0 \quad \text{for any $f\in H'$,} \end{equation} hence I have the weak convergence.

For the inverse implication is enough to evaluate $\| x_{n} -x\|_{H}^{2}$.

But I have no idea how to prove the first point. Someone could help me?

Answer 1

(1a) Let $x_n \rightharpoonup x$. Let $F=\{f_1\dots f_k\}\subset H'$ be finite, $\epsilon>0$. Define $$ U_{F,\epsilon}:=\{x\in H: \ |f_i(x) |< \epsilon \quad \forall i=1\dots k\}. $$ Due to the definition of weak convergence, there is $N$ such that $|f_i(x_n-x)|<\epsilon$ for all $i=1\dots k$, $n>N$. Hence $x_n\in U_{F,\epsilon}$ for all $n>N$. Hence $x_n\to x$ with respect to the weak topology.

(1b) Now let $x_n\to x$ with respect to the weak topology. Take $f\in H'$, $\epsilon>0$. Set $F=\{f\}$. Then there is $N$ such that $x_n \in U_{F,\epsilon}$, which is equivalent to $|f(x-x_n)|<\epsilon$ for all $n>N$. As $\epsilon>0$ was arbitrary, $x_n\rightharpoonup x$ follows.

Answer 2

$\|x_n-x||^{2}=\|x_n\|^{2}+\|x\|^{2}-2 \langle x_n, x \rangle \to \|x\|^{2}+\|x\|^{2}-2\|x\|^{2}=0$. For the last term I have used the fact that $f(y)=\langle y, x \rangle$ defines an element of $H'$. [I have assumed that the scalar field is $\mathbb R$. For the complex case only minor changes are needed and I will let you make these changes].