Exercise 7, § 1

a) Let $F = \overline{\mathscr{K}(\mathbf{N})}$ be the Banach space of sequences of real numbers tending to 0, $F' = L^1(\mathbf{N})$ its dual (TVS, IV, §1, Exer. 1). Let $f = (f_n)$ be the mapping of $I = [0,1]$ into $F$ such that $f_n(t) = n \varphi_{I_n}(t)$, where $I_n = [0,1/n]$. Show that $f$ is scalarly integrable for the Lebesgue measure $\mu$ on $I$, but that $\int f\,d\mu$ is an element of $F''$ not in $F$.

b) Let $g$ be the mapping of $J = [-1,+1]$ into $F$ defined by the conditions $g(t) = f(t)$ for $t \geq 0$, $g(t) = -f(-t)$ for $t \leq 0$. Then $g$ is scalarly integrable for Lebesgue measure on $J$, and $\int g\,d\mu \in F$, but there exist functions $h \in \mathscr{L}^\infty$ such that $\int hg\,d\mu \notin F$.

c) Deduce from a) a new proof of the fact that the topological vector space $F$ is not isomorphic to the dual of a normed space (TVS, IV, §2, Exer. 22 c)).