Let $H$ be a complex Hilbert space with inner product $(\cdot, \cdot)$ and let $T: H \rightarrow H$ be a bounded linear map.

(i) Define the spectrum $\sigma(T)$, the point spectrum $\sigma_{p}(T)$, the continuous spectrum $\sigma_{c}(T)$, and the residual spectrum $\sigma_{r}(T)$.

(ii) Show that $T^{*} T$ is self-adjoint and that $\sigma\left(T^{*} T\right) \subset[0, \infty)$. Show that if $T$ is compact then so is $T^{*} T$.

(iii) Assume that $T$ is compact. Prove that $T$ has a singular value decomposition: for $N<\infty$ or $N=\infty$, there exist orthonormal systems $\left(u_{i}\right)_{i=1}^{N} \subset H$ and $\left(v_{i}\right)_{i=1}^{N} \subset H$ and $\left(\lambda_{i}\right)_{i=1}^{N} \subset[0, \infty)$ such that, for any $x \in H$,

$$T x=\sum_{i=1}^{N} \lambda_{i}\left(u_{i}, x\right) v_{i}$$

[You may use the spectral theorem for compact self-adjoint linear operators.]