(i) Let $T: H_{1} \rightarrow H_{2}$ be a continuous linear map between two Hilbert spaces $H_{1}, H_{2}$. Define the adjoint $T^{*}$ of $T$. Explain what it means to say that $T$ is Hermitian or unitary.

Let $\phi: \mathbb{R} \rightarrow \mathbb{C}$ be a bounded continuous function. Show that the map

$$T: L^{2}(\mathbb{R}) \rightarrow L^{2}(\mathbb{R})$$

with $T f(x)=\phi(x) f(x+1)$ is a continuous linear map and find its adjoint. When is $T$ Hermitian? When is it unitary?

(ii) Let $C$ be a closed, non-empty, convex subset of a real Hilbert space $H$. Show that there exists a unique point $x_{o} \in C$ with minimal norm. Show that $x_{o}$ is characterised by the property

$$\left\langle x_{o}-x, x_{o}\right\rangle \leqslant 0 \quad \text { for all } x \in C .$$

Does this result still hold when $C$ is not closed or when $C$ is not convex? Justify your answers.