(i) State and prove the parallelogram law for Hilbert spaces.

Suppose that $K$ is a closed linear subspace of a Hilbert space $H$ and that $x \in H$. Show that $x$ is orthogonal to $K$ if and only if 0 is the nearest point to $x$ in $K$.

(ii) Suppose that $H$ is a Hilbert space and that $\phi$ is a continuous linear functional on $H$ with $\|\phi\|=1$. Show that there is a sequence $\left(h_{n}\right)$ of unit vectors in $H$ with $\phi\left(h_{n}\right)$ real and $\phi\left(h_{n}\right)>1-1 / n$.

Show that $h_{n}$ converges to a unit vector $h$, and that $\phi(h)=1$.

Show that $h$ is orthogonal to $N$, the null space of $\phi$, and also that $H=N \oplus \operatorname{span}(h)$.

Show that $\phi(k)=\langle k, h\rangle$, for all $k \in H$.