Exercise 18, § 5

a) In the space $\mathbf{R}^N$ of all infinite sequences of real numbers, consider the set J of all sequences $x = (\xi_n)$ for which the number

is finite, the supremum being taken for all integers $m \geq 1$ and for all strictly increasing sequences of integers $p_1 < p_2 < \cdots < p_{m+1}$. Show that $\|x\|$ is a norm on J and that for this norm J is a Banach space. For every $x \in E$, show that the sequence $(\xi_n)$ has a finite limit $u(x)$ and that $u$ is a non zero continuous linear form on E; let $J_0$ denote the closed hyperplane of J with equation $u(x) = 0$ (R. C. James' space). b) Show that the vectors $e_n = (\delta_{mn})_{m \geq 0}$ form a Banach basis of $J_0$ and that this basis is contracting (IV, p. 70, exerc. 15). (To prove the latter point, argue as in exerc. 17, a), by showing that the constructed sequence $(y_k)$, is such that the series with the general term $y_k/k$ is convergent in E; this implies a contradiction.) c) Show that the identity mapping from $J_0$ onto itself can be extended to a topological vector space isomorphism from the strong bidual $J_0''$ onto J, in such a way that $J_0''/J_0$ is 1-dimensional (use IV, p. 70, exerc. 15, b)). Deduce that no Banach basis of $J_0$ can be unconditional (exerc. 17, c)). d) Let $H_1, H_2$ be two closed vector subspaces of $J_0$ generated by the $e_{2n}$ and the $e_{2n+1}$ respectively, for $n \geq 0$. Show that as topological vector spaces, $H_1$ and $H_2$ are isomorphic to the Hilbert space $\ell^2(\mathbf{N})$, and that $J_0$ is not the sum of $H_1$ and $H_2$.*. e) Show that on $J_0$ there exists no complex locally convex space structure having the real locally convex space structure of $J_0$ as the underlying structure (cf. IV, p. 52, exerc. 3).