Exercise 4, § 1

Let T be a set, M an encumbrance on T, E a real or complex Banach space, and p a finite real number such that $p \geq 1$. Denote by $\mathscr{F}$ the set of mappings of T into E, and, for $f$ in $\mathscr{F}$ or in $\mathscr{F}_+(T)$, set $N_p(f) = M(|f|^p)^{1/p}$.

a) For every sequence $(f_n)_{n \geq 1}$ of elements of $\mathscr{F}_+(T)$, $N_p\left( \sum_{n \geq 1} f_n \right) \leq \sum_{n \geq 1} N_p(f_n)$.

b) Let $\mathscr{F}^p$ be the set of $f \in \mathscr{F}$ such that $N_p(f)$ is finite. Show that $\mathscr{F}^p$ is a linear subspace of $\mathscr{F}$, and $N_p$ is a semi-norm on $\mathscr{F}^p$. Show that the set $\mathscr{N}$ of $f \in \mathscr{F}^p$ such that $N_p(f) = 0$ is equal to the set of $f \in \mathscr{F}$ that are zero outside an M-negligible set. Show that the normed space $\mathscr{F}^p / \mathscr{N}$ is complete (cf. Ch. IV, §3, No. 3).