Let $H$ be a Hilbert space and $T \in \mathcal{B}(H)$. Define what is meant by an adjoint of $T$ and prove that it exists, it is linear and bounded, and that it is unique. [You may use the Riesz Representation Theorem without proof.]

What does it mean to say that $T$ is a normal operator? Give an example of a bounded linear map on $\ell_{2}$ that is not normal.

Show that $T$ is normal if and only if $\|T x\|=\left\|T^{*} x\right\|$ for all $x \in H$.

Prove that if $T$ is normal, then $\sigma(T)=\sigma_{\mathrm{ap}}(T)$, that is, that every element of the spectrum of $T$ is an approximate eigenvalue of $T$.