Exercise 20, § 1

Let E be a prehilbert space, let F be a Banach space and let $u: E\rightarrow F$ be a linear mapping.

(a) Prove that $u$ is a compact linear mapping if and only if $\|u(e_n)\| \rightarrow 0$ for every orthonormal sequence $(e_n)$ of E. (To prove that this condition is sufficient, use TG, IX, p. 20, prop. 14.)

(b) Suppose that E is a Hilbert space of countable type and that F = E. For there to exist an orthonormal basis $(e_n)_{n\in\mathbf{N}}$ of E such that $\|u(e_n)\| \rightarrow 0$, it is necessary and sufficient that $u$ not be a Fredholm endomorphism.

(c) Suppose that E is a Hilbert space and that F = E. If $\langle u(e_n)|e_n\rangle \rightarrow 0$ for every orthonormal sequence $(e_n)$ of E, then $u$ is a compact linear mapping.