习题 11, § 3

令$(c_{mn})$为数$> 0$的双序列，使得$c_{m,n} \leq c_{m+1,n}$，并令E为实数的所有序列$x = (x_n)$的空间，使得每整数米有$p_m(x) = \sum_n c_{mn} |x_n| < +\infty$。 我们赋予 E 由半范数 $p_m$ 定义的拓扑，因此拓扑 E 是 Fréchet 空间满足，可数性第一公理； E 的对偶 $E'$ 可以用所有序列 $x' = (x'_n)$ 的空间来识别，使得 $\sup_n c_{mn}^{-1} |x'_n| < +\infty$ 至少 1 m，规范的双线性形式 $\langle x, x' \rangle$ 可以用 $\sum_n x_n x'_n$ 来识别（IV，p. 47，练习 1， c））。 假设不存在任何子序列$(n_k)$，其中存在数$\geq 0$和指标$m_0$的序列$(a_m)$，使得$c_{m,n_k} \leq a_m c_{m_0,n_k}$对于所有$m \geq m_0$和所有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

对于每个指标 q。然后通过 reductio ad absurdum 证明，对于每个 q，至少存在一个指标 $s_q$，使得 $r_q < s_q \leq r_{q+1}$ 并且对于每整数 m，我们有 $c_{m,s_q} \leq b_m 2^{m+2}/\delta$，其中与……矛盾假设。）