Exercise 28, § 1

Let E and F be locally convex spaces and $u\in \mathscr{L}(E; F)$.

a) Prove that the following properties are equivalent:

(i) The image by $u$ of every bounded subset of E is relatively compact for the weak topology ;

(ii) The mapping $^{tt}u$ maps $F''$ into E.

b) If F is quasi-complete, prove that these properties are equivalent to:

(iii) The mapping $^tu$ transforms every equicontinuous subset of $F'$ into a relatively compact subset of $E'$ for $\sigma (E',E'')$. c) If E and F are Banach spaces, (i) to (iii) are equivalent also to

(i’) The mapping $u$ is weakly compact ;

(iii’) The mapping $^tu: F'_b\rightarrow E'_b$ is weakly compact.