Exercise 6, § 2

Let E be a pre-Hilbert 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) Assume that E is a separable Hilbert space and that F = E. For there to exist an orthonormal basis $(e_n)$ of E such that $\|u(e_n)\| \rightarrow 0$, it is necessary and sufficient that $u$ should not be a Fredholm operator.

c) Assume that E is a Hilbert space and that one has $\langle u(e_n), e_n\rangle \rightarrow 0$ for every orthonormal sequence $(e_n)$ of H. Prove that $u$ is a compact linear mapping.