(a) Let $X$ be a separable normed space. For any sequence $\left(f_{n}\right)_{n \in \mathbb{N}} \subset X^{*}$ with $\left\|f_{n}\right\| \leqslant 1$ for all $n$, show that there is $f \in X^{*}$ and a subsequence $\Lambda \subset \mathbb{N}$ such that $f_{n}(x) \rightarrow f(x)$ for all $x \in X$ as $n \in \Lambda, n \rightarrow \infty$. [You may use without proof the fact that $X^{*}$ is complete and that any bounded linear map $f: D \rightarrow \mathbb{R}$, where $D \subset X$ is a dense linear subspace, can be extended uniquely to an element $f \in X^{*}$.]

(b) Let $H$ be a Hilbert space and $U: H \rightarrow H$ a unitary map. Let

$$I=\{x \in H: U x=x\}, \quad W=\{U x-x: x \in H\}$$

Prove that $I$ and $W$ are orthogonal, $H=I \oplus \bar{W}$, and that for every $x \in H$,

$$\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{i=0}^{n-1} U^{i} x=P x$$

where $P$ is the orthogonal projection onto the closed subspace $I$.

(c) Let $T: C\left(S^{1}\right) \rightarrow C\left(S^{1}\right)$ be a linear map, where $S^{1}=\left\{e^{i \theta} \in \mathbb{C}: \theta \in \mathbb{R}\right\}$ is the unit circle, induced by a homeomorphism $\tau: S^{1} \rightarrow S^{1}$ by $(T f) e^{i \theta}=f\left(\tau\left(e^{i \theta}\right)\right)$. Prove that there exists $\mu \in C\left(S^{1}\right)^{*}$ with $\mu\left(1_{S^{1}}\right)=1$ such that $\mu(T f)=\mu(f)$ for all $f \in C\left(S^{1}\right)$. (Here $1_{S^{1}}$ denotes the function on $S^{1}$ which returns 1 identically.) If $T$ is not the identity map, does it follow that $\mu$ as above is necessarily unique? Justify your answer.