Exercise 14, § 5

Let $E$ be a Banach space satisfying the first axiom of countability. A sequence $(e_n)_{n \geq 0}$ of elements of $E$ is said to be a Banach basis if the following property is satisfied : for every $x \in E$, there exists a unique sequence $(a_n)$ of scalars such that $x = \sum_{n=0}^\infty a_n e_n$, where the series on the right hand side is convergent. a) Show that the family $(e_n)$ is total and free. Let $E_n$ be the vector subspace (closed) of $E$ generated by the $e_m$ for indices $m \leq n$, and let $P_n$ be the projector from $E$ onto $E_n$ defined by $P_n \cdot (\sum_{m=0}^\infty a_m e_m) = \sum_{m=0}^n a_m e_m$. Show that the $P_n$ are continuous linear mappings and that $\sup_n \|P_n\| < +\infty$. (Consider the norm on $E$ defined by $\left\| \sum_{n=0}^\infty a_n e_n \right\| = \sup_n \left\| \sum_{m=0}^n a_m e_m \right\|$; show that $E$ is complete for this norm and deduce that it is equivalent to the given norm on $E$ (cf. I, p. 17, th. 1).) Show that for every pair of integers $p < q$, the norm of the projection $P_{q,p}$ of $E_q$ onto $E_p$, which is parallel to the vector subspace generated by the $e_m$ for indices $m$ such that $p + 1 \leq m \leq q$, is bounded by a number independent of $p, q$. b) Conversely, let $(e_n)$ be a total sequence of elements of $E$, which is a free family and is such that the norms of the projections $P_{q,p}$ for $0 \leq p < q$ are bounded by a number $M$ independent of $p$ and $q$. Show that $(e_n)$ is a Banach basis of $E$. (First prove that the sequence $(e_n)$ is topologically free and define the projectors $P_n$; then $\|P_n\| \leq M$ for all $n$. Next observe that if $d(x, E_n)$ is the distance of a point $x \in E$ from $E_n$, then $\|x - P_n \cdot x\| \leq (M + 1)\ d(x, E_n)$.)[^1]

[^1]: It is clear that the existence of a Banach basis in $E$ implies that $E$ satisfies the first axiom of countability. But there are examples of Banach spaces satisfying the first axiom of countability in which there does not exist a Banach basis (P. Enflo, Acta Math., t. CXXX (1973), p. 309-317).