Suppose that $\left(e_{n}\right)$ and $\left(f_{m}\right)$ are orthonormal bases of a Hilbert space $H$ and that $T \in L(H)$.

(a) Show that $\sum_{n=1}^{\infty}\left\|T\left(e_{n}\right)\right\|^{2}=\sum_{m=1}^{\infty}\left\|T^{*}\left(f_{m}\right)\right\|^{2}$.

(b) Show that $\sum_{n=1}^{\infty}\left\|T\left(e_{n}\right)\right\|^{2}=\sum_{m=1}^{\infty}\left\|T\left(f_{m}\right)\right\|^{2}$.

$T \in L(H)$ is a Hilbert-Schmidt operator if $\sum_{n=1}^{\infty}\left\|T\left(e_{n}\right)\right\|^{2}<\infty$ for some (and hence every) orthonormal basis $\left(e_{n}\right)$.

(c) Show that the set HS of Hilbert-Schmidt operators forms a linear subspace of $L(H)$, and that $\langle T, S\rangle=\sum_{n=1}^{\infty}\left\langle T\left(e_{n}\right), S\left(e_{n}\right)\right\rangle$ is an inner product on $H S$; show that this inner product does not depend on the choice of the orthonormal basis $\left(e_{n}\right)$.

(d) Let $\|T\|_{H S}$ be the corresponding norm. Show that $\|T\| \leqslant\|T\|_{H S}$, and show that a Hilbert-Schmidt operator is compact.