(a) State and prove the Riesz representation theorem for a real Hilbert space $H$.

[You may use that if $H$ is a real Hilbert space and $Y \subset H$ is a closed subspace, then $\left.H=Y \oplus Y_{.}^{\perp} .\right]$

(b) Let $H$ be a real Hilbert space and $T: H \rightarrow H$ a bounded linear operator. Show that $T$ is invertible if and only if both $T$ and $T^{*}$ are bounded below. [Recall that an operator $S: H \rightarrow H$ is bounded below if there is $c>0$ such that $\|S x\| \geqslant c\|x\|$ for all $x \in H$.]

(c) Consider the complex Hilbert space of two-sided sequences,

$$X=\left\{\left(x_{n}\right)_{n \in \mathbb{Z}}: x_{n} \in \mathbb{C}, \sum_{n \in \mathbb{Z}}\left|x_{n}\right|^{2}<\infty\right\}$$

with norm $\|x\|=\left(\sum_{n}\left|x_{n}\right|^{2}\right)^{1 / 2}$. Define $T: X \rightarrow X$ by $(T x)_{n}=x_{n+1}$. Show that $T$ is unitary and find the point spectrum and the approximate point spectrum of $T$.