Exercise 14, § 1

Let E be an infinite dimensional Banach space. a) Show that for every integer N, there exists a sequence $(b_j)_{1 \leq j \leq N}$ of N vectors in E, of norm 1, such that, for every sequence $(\xi_j)_{1 \leq j \leq N}$ of N scalars, we have

(use exerc. 12, b)). b) For every sequence $(\lambda_n)_{n \geq 1}$ of numbers $\geq 0$ such that $\sum \lambda_n^2 < +\infty$, show that there exists a sequence $(x_n)_{n \geq 1}$ of points of E such that $\|x_n\| = \lambda_n$ for all $n$, and that the series $(x_n)$ is summable. (Use a) of exerc. 13, a). c) Deduce from b) that in every infinite dimensional Banach space, there exists a commutatively convergent series, that is not absolutely convergent (Dvoretzky-Rogers th.).