Exercise 16, § 5

a) Let E be a Banach space. For an infinite sequence $(x_n)$ of points of E, the following properties are equivalent : α) The series with the general term $x_n$ is commutatively convergent (GT, III, § 5, No. 7). β) For every subset I of $\mathbf{N}$, the series defined by the sequence $(x_n)_{n \in I}$ is convergent (GT, III, § 5, No. 3, prop. 2 and § 5, exerc. 4). γ) 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. δ) For every $\varepsilon > 0$, there exists a finite subset J of $\mathbf{N}$ such that, for every finite subset H of $\mathbf{N}$ not intersecting J, we have $\left\| \sum_{n \in H} x_n \right\| \leq \varepsilon$.

b) Let $(e_n)$ be a Banach basis of E. The following properties are equivalent : α) For every permutation $\pi$ of $\mathbf{N}$, the sequence $(e_{\pi(n)})$ is a Banach basis of E. β) For every sequence $(\varepsilon_n)$ of numbers equal to 1 or to $-1$, the sequence $(\varepsilon_n e_n)$ is a Banach basis of E. γ) For every $x = \sum_{n=0}^\infty \xi_n e_n$ in E and every sequence $(\eta_n)_{n \in \mathbf{N}}$ for which $|\eta_n| \leq |\xi_n|$ for all $n$, the series with the general term $\eta_n e_n$ converges in E. δ) For every $x = \sum_{n=0}^\infty \xi_n e_n$ in E and every strictly increasing sequence $(n_k)_{k \in \mathbf{N}}$ of integers $\geq 0$, the series with the general term $\xi_{n_k} e_{n_k}$ converges in E. ε) There exists a real number $M > 0$ such that, for every finite subset J of $\mathbf{N}$ and for every $x = \sum_{n=0}^\infty \xi_n e_n$ in E, we have $\left\| \sum_{n \in J} \xi_n e_n \right\| \leq M \|x\|$.

(To prove that α) implies ε), argue as in IV, p. 69, exerc. 14. To prove that β) implies γ), reduce to the case where the $\xi_n$ and the $\eta_n$ are real, and consider the sums $\sum_{n=p}^q \langle \eta_n e_n, x' \rangle$ for $x' \in E'$.)

When these conditions are satisfied, $(e_n)$ is said to be an unconditional Banach basis of E.

c) Suppose $(e_n)$ is an unconditional basis. Show that there exists a real number $K > 0$ such that, for every sequence $(\varepsilon_n)$ of numbers equal to 1 or $-1$ and for all $x = \sum_{n=0}^\infty \xi_n e_n$ in E, we have $\left\| \sum_{n=0}^\infty \varepsilon_n \xi_n e_n \right\| \leq M \|x\|$ (same method as in IV, p. 69, exerc. 14). Deduce that for every bounded sequence of scalars $(\lambda_n)$ and for all $x = \sum_{n=0}^\infty \xi_n e_n$ in X, we have

(argue as in b), for the proof that $\beta$ implies $\gamma$).