Exercise 21, § 2

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

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

(For every system $(a_i)_{1 \leq i \leq n}$ of vectors in $F$ such that $|a_i| = 1$, consider the linear form $(\xi_i) \mapsto u(\sum_i \xi_i a_i \varphi_{A_i})$ on $\mathbf{R}^n$, and apply Th. 4 of Ch. V, §5, No. 8 to a measure with finite support.) From this, deduce that if $A = \bigcup_i A_i$, then $\sum_i |m(\varphi_{A_i})| \leq \|u\| (\mu(A))^{1/p}$ (apply Minkowski’s inequality).

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

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

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