Exercise 13, § 1

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

where $(\varepsilon_j)$ ranges over the set of $2^n$ sequences of numbers equal to 1 or to $-1$ (use the identity of the median, cf. V, p. 9, formula (14)). c) Deduce from b) that if $(x_i)_{i \in I}$ is a summable family in a hilbertian space E, the family $(\|x_i\|^2)_{i \in I}$ is summable in $\mathbf{R}$.