Exercise 29, § 1

a) Prove that every weakly compact subset of the Banach space $\ell^1(\mathbf{N})$ is compact (cf. EVT, IV, p. 69, Exercise 11). In particular, every weakly compact linear mapping from a locally convex space E into $\ell^1(\mathbf{N})$ is compact.

b) Let E be a Banach space. Deduce from (a) that every weakly compact linear mapping from $c_0$ into E is compact.