Exercise 13, § 2

a) Let A be an infinite set of unit vectors of E which converges weakly to 0 with respect to the filter of the complement of finite subsets. Prove that there exists a sequence $(x_n)_{n\geqslant 1}$ with values in A and an orthonormal sequence $(e_n)_{n\geqslant 1}$ in E such that $\|x_n-e_n\|\leqslant 1/n$ for all $n\geqslant 1$.

b) Let $u$ be an endomorphism of E, not necessarily continuous. Suppose that for every orthonormal sequence $(e_n)$ in E, the sequence $(u(e_n))$ converges weakly to 0. Prove that $u$ is continuous and compact.

c) Let $u$ be a continuous endomorphism of E. In order that there exist an orthonormal basis $(e_n)$ of E such that the sequence $(u(e_n))$ converges weakly to 0, it is necessary and sufficient that for every endomorphism $v$ of $E,v\circ u-1_E$ be not compact.