Exercise 4, § 4

Let E be a Banach space in which there exists an infinite algebraically independent sequence $(a_n)$ which is total in E (for example the space $\ell^1(\mathbf{N})$ (I, p. 4)). Let B be a base of E containing the $a_n$; we know (II, p. 80, exerc. 24) that B is not countable. Let $(e_n)$ be a sequence of distinct elements of B, and distinct from the $a_n$, and let C be the complement of the set of the $e_n$ in B. Let $F_n$ be the vector subspace of E generated by C and the $e_k$ for indices $k \leq n$; E is the union of the $F_n$. Let S be the unit ball in E; show that there exists an index n such that $S \cap F_n$, is not a first category set. Deduce that for this value of n, $F_n$ is a metrizable, non-complete Baire space.