State and prove the Riesz representation theorem for bounded linear functionals on a Hilbert space $H$.

[You may assume, without proof, that $H=E \oplus E^{\perp}$, for every closed subspace $E$ of $H$.]

Prove that, for every $T \in \mathcal{B}(H)$, there is a unique $T^{*} \in \mathcal{B}(H)$ such that $\langle T x, y\rangle=\left\langle x, T^{*} y\right\rangle$ for every $x, y \in H$. Prove that $\left\|T^{*} T\right\|=\|T\|^{2}$ for every $T \in \mathcal{B}(H)$.

Define a normal operator $T \in \mathcal{B}(H)$. Prove that $T$ is normal if and only if $\|T x\|=\left\|T^{*} x\right\|$ for every $x \in H$. Deduce that every point in the spectrum of a normal operator $T$ is an approximate eigenvalue of $T$.

[You may assume, without proof, any general criterion for the invertibility of a bounded linear operator on $H$.]