(i) Define the adjoint of a bounded, linear map $u: H \rightarrow H$ on the Hilbert space $H$. Find the adjoint of the map

$$u: H \rightarrow H ; \quad x \mapsto \phi(x) a$$

where $a, b \in H$ and $\phi \in H^{*}$ is the linear map $x \mapsto\langle b, x\rangle$.

Now let $J$ be an incomplete inner product space and $u: J \rightarrow J$ a bounded, linear map. Is it always true that there is an adjoint $u^{*}: J \rightarrow J$ ?

(ii) Let $\mathcal{H}$ be the space of analytic functions $f: \mathbb{D} \rightarrow \mathbb{C}$ on the unit disc $\mathbb{D}$ for which

$$\iint_{\mathbb{D}}|f(z)|^{2} d x d y<\infty \quad(z=x+i y)$$

You may assume that this is a Hilbert space for the inner product:

$$\langle f, g\rangle=\iint_{\mathbb{D}} \overline{f(z)} g(z) d x d y .$$

Show that the functions $u_{k}: z \mapsto \alpha_{k} z^{k}(k=0,1,2, \ldots)$ form an orthonormal sequence in $\mathcal{H}$ when the constants $\alpha_{k}$ are chosen appropriately.

Prove carefully that every function $f \in \mathcal{H}$ can be written as the sum of a convergent series $\sum_{k=0}^{\infty} f_{k} u_{k}$ in $\mathcal{H}$ with $f_{k} \in \mathbb{C}$.

For each smooth curve $\gamma$ in the disc $\mathbb{D}$ starting from 0 , prove that

$$\phi: \mathcal{H} \rightarrow \mathbb{C} ; \quad f \mapsto \int_{\gamma} f(z) d z$$

is a continuous, linear map. Show that the norm of $\phi$ satisfies

$$\|\phi\|^{2}=\frac{1}{\pi} \log \left(\frac{1}{1-|w|^{2}}\right)$$

where $w$ is the endpoint of $\gamma$.