Exercise 5, § 3

For every integer $n > 0$, let $a^{(n)}$ be the double sequence $(a_{pq}^{(n)})$ ($p \in \mathbf{N}, q \in \mathbf{N}$) such that $a_{pq}^{(n)} = q$ if $p \leq n$ and $a_{pq}^{(n)} = 1$ if $p > n$. Let E be the vector space of all double sequences $x = (x_{pq})_{(p,q) \in \mathbf{N} \times \mathbf{N}}$ of real numbers such that, for every integer $n > 0$, the number $r_n(x) = \sum_{p,q} a_{pq}^{(n)} |x_{pq}|$ is finite. If E is assigned the topology defined by the semi-norms $r_n$, then E is a Fréchet space satisfying the first axiom of countability (IV, p. 47, exerc. 1, c)); the dual E' of E can be identified with the space of all double sequences $x' = (x'_{pq})$ of real numbers such that for at least one index $n$, there exists $k_n > 0$ such that $|x'_{pq}| \leq k_n a_{pq}^{(n)}$ for every pair $(p, q)$; and $\langle x, x' \rangle = \sum_{p,q} x_{pq} x'_{pq}$ (IV, p. 47, exerc. 1, c)). For every integer $p_0 > 0$ and every sequence $(m_p)$ of integers $> 0$, let $J(p_0; (m_p))$ be the set of pairs of integers $p > 0, q > 0$ such that $p \geq p_0$ and $q \geq m_p$; let $\mathscr{B}$ be the filter base on $\mathbf{N} \times \mathbf{N}$ consisting of the sets $J(p_0; (m_p))$ and let $\mathfrak{F}$ be an ultrafilter which is finer than the filter with base $\mathscr{B}$. a) Show that for all $x' = (x'_{pq}) \in E'$, the double sequence $(x'_{pq})$ has a limit $u(x')$ with respect to the ultrafilter $\mathfrak{F}$; if $V_n$ is a neighbourhood of 0 in E defined by $r_n(x) \leq 1$, then $|u(x')| \leq 1$ for all $x' \in V_n^\circ$. b) Let $U'$ be a neighbourhood of 0 in $E'$, for the strong topology, which is convex, balanced and weakly closed, and for every $n$, let $\alpha_n > 0$ be such that $\alpha_n V_n^\circ \subset U'$. For every integer $p > 0$, let $m_p$ be an integer such that $2^{p+1} \leq \alpha_p m_p$, and let $x' = (x'_{pq})$ be the double sequence with $x'_{pq} = 0$ for $q < m_p$, $x'_{pq} = 2$ for $q \geq m_p$. Show that $x' \in U'$ but that $u(x') = 2$; deduce that $u$ is not strongly continuous in $E'$, while being bounded on every bounded subset of $E'$. Conclude (IV, p. 58, exerc. 3) that E is not distinguished, and consequently that the strong dual $E'_b$ is a non infra-barrelled (DF) space. c) Using b) construct an example of a closed subspace M of a Fréchet space F such that the strong topology $\beta(F'/M^\circ, M)$ is distinct from the quotient topology by $M^\circ$ of the strong topology $\beta(F', F)$ (embed E in a countable product of Banach spaces).