Exercise 13, § 1

Let E be a complex Hilbert space of countable type, $(e_n)_{n\in\mathbf{N}}$ an orthonormal basis of E and $u$ an endomorphism of E. Now prove that $u$ is compact if and only if one has

lim$_{n\rightarrow+\infty x\in \{}$sup$_{e_0,...,e_{n-1}\}^{\circ}-\{0\}}\frac{\|u(x)\|^2}{\|x\|^2}= 0$.