Exercise 15, § 5

Let E be the space of all sequences x = (\xi_n)_{n \geq 0} of real numbers such that the series with the general term \xi_n is convergent. Put \| x \| = \sup_n | \sum_{k=0}^n \xi_k |.

a) Show that \| x \| is a norm on E, and that E is complete for this norm. b) Show that the vector space \ell^1(\mathbf{N}) (I, p. 4), considered as a subspace of E, is everywhere dense (for the topology of E); the topology on \ell^1(\mathbf{N}) defined by the norm $\| x \|_1 = \sum_{n=0}^\infty | \xi_n |$ is strictly finer than the topology induced by that of E. c) Let (P_n) be an increasing sequence of finite subsets of \mathbf{N} \times \mathbf{N} forming a cover of \mathbf{N} \times \mathbf{N}. For every x = (\xi_n) \in E and every y = (\eta_n) \in \ell^1(\mathbf{N}), let $f_n(x, y) = \sum_{(i,j) \in P_n} \xi_i \eta_j$. Then the sequence (f_n(x, y)) tends to a limit for every pair (x, y) \in E \times \ell^1(\mathbf{N}) if and only if for each of these pairs (x, y), the sequence (f_n(x, y)) is bounded; the limit of f_n(x, y) is then equal to $(\sum_{n=0}^\infty \xi_n) (\sum_{n=0}^\infty \eta_n)$.

(Using exerc. 13, c) of III, p. 48, show that the sequence of bilinear forms $(f_n)$ is equicontinuous, and observe that it converges in the subspace $l^1(\mathbf{N}) \times l^1(\mathbf{N})$; conclude the argument using b).) d) For every $j \in \mathbf{N}$, let $\rho_{jn}$ be the smallest number of closed intervals of $\mathbf{N}$ whose union is the projection of $P_n \cap (\mathbf{N} \times \{j\})$ onto $\mathbf{N}$; let $\rho_n = \sup_{j \in \mathbf{N}} \rho_{jn}$. Show that the condition obtained in c) is equivalent to $\sup_n \rho_n < +\infty$. (If $\phi_n$ is the characteristic function of $P_n$, show that the norm of the bilinear form $f_n$ is $\sup_{j \in \mathbf{N}} (\sum_{i=0}^\infty |\rho_n(i, j) - \phi_n(i+1, j)|)$.)