Exercise 30, § 1

a) Let E and F be Banach spaces and $u\in \mathscr{L}(E; F)$. Now prove that the following properties are equivalent:

(i) The image by $u$ of a weakly compact subset of E is (strongly) compact ;

(ii) The image by $u$ of a weakly relatively compact subset of E is relatively compact in F ;

(iii) The image by $u$ of a weakly convergent sequence in E converges strongly in F.

(Use the theorem of Šmulian, EVT, IV, p. 36, th. 2.)

One will say that $u$ is semi-compact[^1] if it satisfies these properties.

b) Let E and F be Banach spaces. Exercise prove that the set $\mathscr{L}^{sc}(E; F)$ of semi-compact mappings is a closed subspace of $\mathscr{L}(E; F)$.

c) Let E and F be Banach spaces and $u\in \mathscr{L}(E; F)$. Let $E_1, F_1$ be Banach spaces, $f\in \mathscr{L}(E_1; E)$ and $g\in \mathscr{L}(F; F_1)$. If $u: E\rightarrow F$ is semi-compact, the same is true of $g\circ u\circ f$.

d) Let E and F be Banach spaces and $u\in \mathscr{L}(E; F)$. If $u$ is compact, it is semi-compact. Conversely, if E is reflexive and $u$ semi-compact, then $u$ is compact.

e) Let E and F be Banach spaces. If the space $E'$ dual of E is of countable type then every completely continuous linear mapping $E\rightarrow F$ is compact.

f) Let I be an infinite set ; the identity mapping of $\ell^1(I)$ is semi-compact, but is not weakly compact (cf. EVT, IV, p. 54, exerc. 15). g) The injection of $\ell^1_K(\mathbf{N})$ into $\ell^2_K(\mathbf{N})$ is semi-compact but is not compact (cf. loc. cit.).

h) Let E be a Banach space. Every continuous linear mapping $E\rightarrow$ $\ell^1_K(\mathbf{N})$ is semi-compact, and every continuous linear mapping $\ell^1_K(\mathbf{N})\rightarrow E$ is semi-compact. i) Let E be a Banach space. If E is reflexive or if its dual is of countable type, then every continuous linear mapping $E\rightarrow \ell^1_K(\mathbf{N})$ is compact.

[^1]: Some authors speak of completely continuous mappings.