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 all sequences $x = (x_n)$ of real numbers such that $p_m(x) = \sum_n c_{mn} |x_n| < +\infty$ for every integer m. We endow E with the topology defined by the semi-norms $p_m$ and for this topology E is a Fréchet space satisfying the first axiom of countability; the dual $E'$ of E can be identified with the space of all 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, exerc. 1, c)). Suppose that there does not exist any subsequence $(n_k)$ for which there is a sequence $(a_m)$ of numbers $\geq 0$ and an index $m_0$ such that $c_{m,n_k} \leq a_m c_{m_0,n_k}$ for all $m \geq m_0$ and for all k. Under these conditions, every weakly convergent sequence in $E'$ is strongly convergent and consequently (IV, p. 60, exerc. 10) E is a Montel space. (Argue by reductio ad absurdum; if necessary make a transformation of the form $(x_n) \mapsto (a_n x_n)$ to reduce to the case where $c_{m_0 n} = 1$ for all n and some $m_0$, and where there exists a sequence $(x^{(p)})_{p \geq 0}$ which converges weakly to 0 in E', and is such that $|x^{(p)}_n| \leq 1$ for every pair $(p, n)$, and also that there exists 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. Under these hypothesis, prove 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. Then show, by reductio ad absurdum, 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, we have $c_{m,s_q} \leq b_m 2^{m+2}/\delta$, which contradicts the hypothesis.)