Exercise 17, § 5

a) Let E be a Banach space with an unconditional Banach basis $(e_n)$ (exerc. 16). Show that if the basis $(e_n)$ is not contracting (IV, p. 70, exerc. 15), then there exists a number $\alpha > 0$, a linear form $x' \in E'$ such that $\|x'\| = 1$, a strictly increasing sequence of integers $(n_k)$ and for each $k$, an element $y_k$ which is a linear combination of the $e_j$ for $n_k \leq j \leq n_{k+1}$, and is such that $\|y_k\| \leq 1$ and $\langle y_k, x' \rangle \geq \alpha$. Deduce that for every finite sequence $(\lambda_j)_{1 \leq j \leq m}$ of scalars, we have $\left\| \sum_{j=1}^m \lambda_j y_j \right\| \geq \frac{\alpha}{2K} \sum_{j=1}^m |\lambda_j|$ (use exerc. 16, c)). Conclude that there exists a topological vector space isomorphism from $\ell^1(\mathbf{N})$ onto a closed subspace of E. b) Deduce from a) that if the strong dual $E'$ of E satisfies the first axiom of countability, then every unconditional basis $(e_n)$ of E is contracting, and hence $(e'_n)$ is an unconditional basis of $E'$ (IV, p. 70, exerc. 15, a)). (Observe that if a closed vector subspace of E is isomorphic to $\ell^1(\mathbf{N})$, then E' cannot satisfy the first axiom of countability.) c) Show that if E has an unconditional basis and if the strong bidual $E''$ of E satisfies the first axiom of countability, then E is reflexive. (Using IV, p. 51, exerc. 25, note that the strong dual $E'$ of E satisfies the first axiom of countability; then use IV, p. 70, exerc. 15, c) and IV, p. 53, exerc. 11.)