Exercise 11, § 3

Let $(c_{mn})$ be a double sequence of numbers $> 0$ such that $c_{m,n} \leq c_{m+1,n}$ and let $E$ be the space of sequences $x = (x_n)$ of real numbers such that one has $p_m(x) = \sum_n c_{mn}|x_n| < + \infty$ for every integer $m$. We endow $E$ with the topology defined by the seminorms $p_m$, for which $E$ is a Fréchet space of countable type; the dual $E'$ of $E$ is identified with the space of sequences $x' = (x'_n)$ such that $\sup_n c_{mn}^{-1}|x'_n| < + \infty$ for at least one $m$, the canonical bilinear form $\langle x, x' \rangle$ being identified with $\sum_n x_n x'_n$ (IV, p. 47, Exercise 1, c)). It is assumed that there is no subsequence $(n_k)$ such that there are a sequence $(a_m)$ of numbers $\geq 0$ and an index $m_0$ for which one has $c_{m,n_k} \leq a_m c_{m_0,n_k}$ for all $m \geq m_0$ and $k$. Under these conditions, every weakly convergent sequence in $E'$ is strongly convergent, and consequently (IV, p. 60, Exercise 10) $E$ is a Montel space. (Reason by contradiction; by making, if necessary, a transform of the form $(x_n) \mapsto (a_n x_n)$, reduce to the case where $c_{m_0 n} = 1$ for all $n$ and an index $m_0$, and where there exists a sequence $(x^{(p)})_{p \geq 0}$ weakly convergent to 0 in $E'$, such that $|x^{(p)}_n| \leq 1$ for every pair $(p, n)$, and a bounded set $B$ in $E$, defined by $p_m(x) \leq b_m$ for all $m \geq 0$, and such that $\sup_{x \in B} |\langle x, x^{(p)} \rangle| \geq 2\delta > 0$ for every integer $p$. Deduce from these assumptions that there exists a strictly increasing sequence $(r_q)$ of integers, and a sequence $(x^{(q)})$ of points of $B$ such that

for each index $q$. Now prove, by reasoning by contradiction, that for every $q$, there exists at least one index $s_q$ such that $r_q < s_q \leq r_{q+1}$ and that for every integer $m$, one has $c_{m,s_q} \leq b_m 2^{m+2}/\delta$, a result contrary to the assumption.)