Exercise 21, § 1

Let E be a Hilbert space of countable type.

(a) Let A be an infinite set of unit vectors of E which converges weakly to 0 according 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 a linear mapping of E into 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 an endomorphism of E. There exists an orthonormal basis $(e_n)_{n\in\mathbf{N}}$ of E such that the sequence $(u(e_n))_{n\in\mathbf{N}}$ converges weakly to 0 if and only if there does not exist an endomorphism $v$ of E such that $v\circ u-1_E$ is compact.