Let $H$ be a separable Hilbert space and $\left\{e_{i}\right\}$ be a Hilbertian (orthonormal) basis of $H$. Given a sequence $\left(x_{n}\right)$ of elements of $H$ and $x_{\infty} \in H$, we say that $x_{n}$ weakly converges to $x_{\infty}$, denoted $x_{n} \rightarrow x_{\infty}$, if $\forall h \in H, \lim _{n \rightarrow \infty}\left\langle x_{n}, h\right\rangle=\left\langle x_{\infty}, h\right\rangle$.

(a) Given a sequence $\left(x_{n}\right)$ of elements of $H$, prove that the following two statements are equivalent:

(i) $\exists x_{\infty} \in H$ such that $x_{n} \rightarrow x_{\infty}$;

(ii) the sequence $\left(x_{n}\right)$ is bounded in $H$ and $\forall i \geqslant 1$, the sequence $\left(\left\langle x_{n}, e_{i}\right\rangle\right)$ is convergent.

(b) Let $\left(x_{n}\right)$ be a bounded sequence of elements of $H$. Show that there exists $x_{\infty} \in H$ and a subsequence $\left(x_{\phi(n)}\right)$ such that $x_{\phi(n)} \rightarrow x_{\infty}$ in $H$.

(c) Let $\left(x_{n}\right)$ be a sequence of elements of $H$ and $x_{\infty} \in H$ be such that $x_{n} \rightarrow x_{\infty}$. Show that the following three statements are equivalent:

(i) $\lim _{n \rightarrow \infty}\left\|x_{n}-x_{\infty}\right\|=0$;

(ii) $\lim _{n \rightarrow \infty}\left\|x_{n}\right\|=\left\|x_{\infty}\right\|$;

(iii) $\forall \epsilon>0, \exists I(\epsilon)$ such that $\forall n \geqslant 1, \sum_{i \geqslant I(\epsilon)}\left|\left\langle x_{n}, e_{i}\right\rangle\right|^{2}<\epsilon$.