Exercise 14, § 2

a) Let $(E_n)_{n \geq 0}$ be an infinite sequence of real Banach spaces, E the vector subspace of the product $F = \prod_{n=0}^\infty E_n$ consisting of all sequences $x = (x_n)$ such that $\sum_n \|x_n\|^2 < +\infty$. Show that the function $\|x\| = (\sum_n \|x_n\|^2)^{1/2}$ on E, is a norm, and that E is complete for this norm : we say that E is the hilbertian sum of the Banach spaces $E_n$. b) Show that the strong dual $E'$ of E can be identified with the hilbertian sum of the strong duals $E'_n$ of the spaces $E_n$, and that if $x' = (x'_n) \in E'$, then $\langle x, x' \rangle = \sum_n \langle x_n, x'_n \rangle$ (if u is a continuous linear form on E, $u_n$ its restriction to $E_n$ considered as a subspace of E, and $a_n$ a point of $E_n$ such that $\|a_n\| = 1$, show that, for every sequence $(\lambda_n)$ of real numbers such that $\sum \lambda_n^2 < +\infty$ the series with the general term $\lambda_n u_n(a_n)$ is convergent, and deduce that $\sum_n (u_n(a_n))^2 < +\infty$, using Banach-Steinhaus theorem for $\ell^2(\mathbf{N})$, for example). c) Deduce from b) that when each of the $E_n$ is reflexive, then E is reflexive. In particular, if for $E_n$ we take the space $\mathbf{R}^n$ endowed with the norm $\|x\| = \sup_{1 \leq i \leq n} |\xi_i|$ for $x = (\xi_i)_{1 \leq i \leq n}$, show that E is reflexive, but there does not exist any norm compatible with the topology of E and for which E is uniformly convex (V, p. 67, exerc. 31).