Let $H$ be a Hilbert space. Show that if $V$ is a closed subspace of $H$ then any $f \in H$ can be written as $f=v+w$ with $v \in V$ and $w \perp V$.

Suppose $U: H \rightarrow H$ is unitary (that is to say $U U^{*}=U^{*} U=I$ ). Let

$$A_{n} f=\frac{1}{n} \sum_{k=0}^{n-1} U^{k} f$$

and consider

$$X=\{g-U g: g \in H\}$$

(i) Show that $U$ is an isometry and $\left\|A_{n}\right\| \leqslant 1$.

(ii) Show that $X$ is a subspace of $H$ and $A_{n} f \rightarrow 0$ as $n \rightarrow \infty$ whenever $f \in X$.

(iii) Let $V$ be the closure of $X$. Show that $A_{n} v \rightarrow 0$ as $n \rightarrow \infty$ whenever $v \in V$.

(iv) Show that, if $w \perp X$, then $U w=w$. Deduce that, if $w \perp V$, then $U w=w$.

(v) If $f \in H$ show that there is a $w \in H$ such that $A_{n} f \rightarrow w$ as $n \rightarrow \infty$.