Exercise 14, § 1

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

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