Exercise 13, § 1

a) Let $(x_n)_{n \in \mathbf{N}}$ be an infinite sequence in a Banach space E. Now prove that, for the family $(x_n)$ to be summable, it is necessary and sufficient that, for every sequence $(\varepsilon_n)$ of numbers equal to 1 or to $-1$, the series of general term $(\varepsilon_n x_n)$ be convergent (use TG, III, p. 79, § 5, exerc. 4). b) Let $(x_j)_{1 \leq j \leq n}$ be a finite sequence of points of a Hilbert space E. Now prove that one has

c) Deduce from b) that if $(x_i)_{i \in I}$ is a summable family in a Hilbert space E, the family $(\|x_i\|^2)_{i \in I}$ is summable in $\mathbf{R}$.