Exercise 26, § 1

Let E and F be Banach spaces and $u$ a continuous linear mapping from E into F.

a) Prove that the following conditions are equivalent:

(i) The mapping $u$ is compact ;

(ii) There exists a sequence $(x'_n)_n$ tending to 0 in the strong dual $E'$ of E satisfying $\|u(x)\|\leqslant$ sup$|\langle x, x'_n\rangle |$ for every $x\in E$;

(iii) There exists a closed subspace L of the Banach space $c_0$ and compact linear mappings $f: E\rightarrow L$ and $g: L\rightarrow F$ such that $u=g\circ f$. (If $u$ is compact, apply Cor. 1 of EVT, IV, p. 24 to the image by $^tu$ of the polar of the unit ball of F. Under assumption (ii), let $(\lambda_n)$ be an element of $c_0$ such that the sequence $(\lambda^{-1}_nx'_n)$ tends to 0 ; define $f$ by $f(x) = (\langle x, \lambda^{-1}_nx'_n\rangle )_n$ and take for L the closure of the image of $f.$)

b) If $u$ is compact, one has $\|u\|=$ inf(sup$_n\|x'_n\|$), the greatest lower bound being taken over the set of sequences $(x'_n)$ satisfying (ii), and $\|u\|=$ inf($\|f\|\|g\|$), the greatest lower bound being taken over the set of triples $(L, f, g)$ satisfying the conditions of (iii).)