Exercise 15, § 2

a) For every integer $n > 0$, let $a^{(n)}$ be the double sequence defined in IV, p. 63, exerc. 8. Let E be the vector space of double sequences $x = (x_{ij})$ of real numbers such that, for every integer $n > 0$, we have $p_n(x) = (\sum_{i,j} a_{ij}^{(n)} |x_{ij}|^2)^{1/2} < + \infty$. Show that the $p_n$ are semi-norms on E, and that E endowed with the topology defined by these semi-norms is a Fréchet space and a Montel space (argue as in IV, p. 60, exerc. 11). b) Show that the dual of E can be identified with the space $E'$ of all double sequences $x' = (x'_{ij})$ such that, for at least one index $n$, we have $\sum_{i,j} (a_{ij}^{(n)})^{-1} |x'_{ij}|^2 < + \infty$. c) For every $x = (x_{ij}) \in E$, show that $\sum_{j=1}^\infty (\sum_{i=1}^\infty |x_{ij}|^2) < + \infty$ (use Cauchy-Schwarz inequality); for every $j \geq 1$, put $y_j = \sum_{i=1}^\infty x_{ij}$; the sequence $u(x) = (y_j)$ then belongs to the hilbertian space $\ell^2(\mathbf{N})$. Show that $u$ is a strict surjective morphism from E onto $\ell^2(\mathbf{N})$; deduce that there exists weakly compact sets in $\ell^2(\mathbf{N})$ which are not the images under $u$ of a bounded set in E (argue as in IV, p. 63, exerc. 8).