Exercise 11, § 1

Let $E$ and $F$ be Banach spaces over K. A linear mapping $u\in \mathscr{L}(E; F)$ is said to be completely continuous if, for every sequence $(x_n)_{n\in N}$ in E, weakly convergent, the sequence $(u(x_n))_{n\in\mathbf{N}}$ converges in F.

a) Every compact linear mapping is completely continuous.

b) If the dual space $E'$ of E is of countable type then every completely continuous linear mapping $E\rightarrow F$ is compact.

c) Let $(u_n)_{n\in\mathbf{N}}$ be a sequence of elements of $\ell^1_K(\mathbf{N})$. Now prove that $(u_n)$ converges if and only if it converges weakly (« Schur theorem »; cf. EVT, IV, p. 54, exercise 15).

d) Deduce that the injection of $\ell^1_K(\mathbf{N})$ into $\ell^2_K(\mathbf{N})$ is completely continuous but is not compact.

e) For every Banach space, every continuous linear mapping $E\rightarrow \ell^1_K(\mathbf{N})$ is completely continuous, and every continuous linear mapping $\ell^1_K(\mathbf{N})\rightarrow E$ is completely continuous.

f ) If E is reflexive or if its dual is of countable type, every continuous linear mapping $E\rightarrow \ell^1_K(\mathbf{N})$ is compact.