Exercise 21, § 2

Let $F$ be a Banach space, $u$ a continuous linear form on the Banach space $L^p_F(T, \mu)$; denote by $q$ the conjugate exponent of $p$, by $\mathscr{E}$ the space of numerical step functions on the $\mu$-integrable sets. One can write, for $f \in \mathscr{L}^p, z \in F, u(fz) = \langle z, m(f) \rangle$, where $m$ is a continuous linear mapping from $\mathscr{L}^p$ into $F'$ such that $|m(f)| \leq \|u\| . N_p(f)$.

a) Let $(A_i)_{1 \leq i \leq n}$ be a finite sequence of non-negligible $\mu$-integrable sets, pairwise disjoint. Prove that one has

(For every system $(\mathbf{a}_i)_{1 \leq i \leq n}$ of vectors of $F$ such that $|\mathbf{a}_i| = 1$, consider the linear form $(\xi_i) \to u(\sum_i \xi_i \mathbf{a}_i \varphi_{A_i})$ on $\mathbf{R}^n$, and apply Theorem 4 of Chapter V, § 5, No. 8, to a measure of finite support.) Deduce that, if $A = \bigcup_i A_i$, one has $\sum_i |\mathbf{m}(\varphi_{A_i})| \leq \|u\| (\mu(A))^{1/p}$ (apply Minkowski's inequality).

b) For every positive function $f \in \mathscr{E}$, put $\nu(f) = \sup (\sum_i |\mathbf{m}(f \varphi_{A_i})|)$ where $(A_i)$ runs through the set of finite sequences of $\mu$-integrable sets, pairwise disjoint. Prove that $\nu$ is the restriction to $\mathscr{E}$ of a positive measure with base $\mu$ (still denoted by $\nu$) on $T$, such that $|\mathbf{m}(f)| \leq \nu(f)$ for every positive function $f \in \mathscr{E}$ (use a).

c) Prove that, if $F$ is of countable type, there exists a mapping $g$ of $T$ into $F'$, scalarly $\mu$-measurable, such that $u(\tilde{f}) = \int \langle \tilde{f}, g \rangle d\mu$ for every function $f \in L_p^p$, and that $\|u\| = N_q(g)$. (To establish the last equality, use Proposition 13 of § 1, No. 5 and Exercise 13 of § 1.)

d) Deduce from c) that, if $F$ is a reflexive Banach space of countable type, the space $L_p^p$ is reflexive for $1 < p < +\infty$ (apply Proposition 12 of § 1, No. 5).