Exercise 31, § 1

Let $E$ be a Banach space.

a) Prove that the following conditions are equivalent:

(i) Whatever the Banach space $F$, every weakly compact mapping from E into F is semi-compact ;

(ii) If $(x_n)$ is a sequence tending to 0 in E for the topology $\sigma (E,E')$ and $(x'_n)$ is a sequence tending to 0 in $E'$ for the topology $\sigma (E',E'')$, the sequence $\langle x_n, x'_n\rangle$ tends to 0.

(To prove that (i) implies (ii), one will take $F =c_0$. If (i) is not satisfied, there exists a weakly compact mapping $u$ from E into a Banach space F and a sequence $(x_n)$ in E tending weakly to 0, such that inf($\|u(x_n)\|$)$>0$; one will construct a sequence $(y_n)$ in the unit ball of $E'$, convergent for $\sigma (E',E'')$ and such that $\langle u(x_n), y_n\rangle =\|u(x_n)\|$, and one will take $x'_n=y_n-$ lim $y_n.$)

These conditions are verified in particular when $E =\mathscr{C}$(T), where T is a compact space, and when $E = L^1(X, \mu)$, or $E = L^{\infty}(X, \mu)$, where X is a locally compact space and $\mu$ a measure on X (INT, VI, § 2, exerc. 23). b) Prove that a Banach space which possesses the above properties admits no direct reflexive factor of infinite dimension.

c) Let H be a reflexive Banach space, and let B be the unit ball of $H'$, equipped with the weak topology. Show that H identifies with a closed non-direct subspace of $\mathscr{C}(B)$.