Exercise 8, § 4

For every integer $n > 0$, let $a^{(n)}$ be the double sequence $a^{(n)} = (a_{ij}^{(n)})_{i \geq 1, j \geq 1}$ defined by $a_{ij}^{(n)} = j^n$ for every pair $(i, j)$ such that $i < n$ and $a_{ij}^{(n)} = i^n$ for every pair $(i, j)$ such that $i \geq n$. Let E be the vector space of double sequences $x = (x_{ij})$ of real numbers such that, for every integer $n > 0$, we have $p_n(x) = \sum_{i,j} a_{ij}^{(n)} |x_{ij}| < + \infty$; the semi-norms $p_n$ define a topology of a Fréchet space and of a Montel space on E (IV, p. 60, exerc. 11); the dual $E'$ of E, which is a (DF) space and a Montel space (hence ultrabornological and reflexive) is identical with the space of all sequences $x' = (x'_{ij})$ such that, for at least one index $n$, we have the relation $\sup_{i,j} |a_{ij}^{(n)}|^{-1} |x'_{ij}| < + \infty$ (IV, p. 47, exerc. 1, $c$).

a) For every $x = (x_{ij}) \in E$, let $y_j = \sum_i x_{ij}$ (for all $j \geq 1$). Show that $\sum_j |y_j| < + \infty$; let $u(x)$ denote the sequence $(y_i) \in \ell^1(\mathbf{N})$; show that $u$ is a continuous linear mapping from E into $F = \ell^1(\mathbf{N})$, and that for every sequence $y' = (y'_j) \in F' = \ell^\infty(\mathbf{N})$, $u'(y')$ is the sequence $(z'_{ij}) \in E'$ for which $z'_{ij} = y'_j$ for every index $i$. Deduce that $u'$ is an injective linear mapping from $\ell^\infty(\mathbf{N})$ onto a subspace of $E'$ which is closed for $\sigma(E', E)$, and consequently that $u$ is a strict morphism from E onto $\ell^1(\mathbf{N})$ for the initial topologies, and $u'$ is an isomorphism from $F' = \ell^\infty(\mathbf{N})$ onto $u'(F')$ for the weak topologies $\sigma(F', F)$ and $\sigma(E', E)$.

b) Let $M = u^{-1}(0)$; then $M^\circ = u'(F')$. Show that the inverse image under $u'$ of the topology induced on $M^\circ$ by the strong topology of $E'$, is the topology of uniform convergence on compact subsets of $F$ (IV, p. 28, th. 1 and p. 54, exerc. 15). Deduce that on $M^\circ$, the topology induced by the strong topology $\beta(E', E)$ is not the strong topology $\beta(M^\circ, E/M)$, and that for the topology induced by $\beta(E', E)$, $M^\circ$ is not an infra-barrelled space, in spite of being a closed subspace of an ultrabornological Montel space; on the other hand, $E/M$, which is the quotient of a Fréchet and a Montel space by a closed subspace, is not reflexive. Show that in $E/M$ there exist bounded sets which are not the canonical images of bounded subsets of E.