Exercise 24, § 5

Let $E$ be an infinite dimensional, metrisable, locally convex space. a) Show that there exists a sequence $(a_n)$ of points of $E$ tending to 0 and a decreasing sequence $(L_n)$, of closed vector subspaces of $E$, such that $L_n$ is of codimension $n$ in $E$ and that, for all $n$, the point $a_n$ belongs to $L_n - L_{n+1}$. b) Suppose further that $E$ is complete. Show that we can then find sequences $(a_n)$ and $(L_n)$ verifying the conditions a), and such that in addition, for every bounded sequence of real number $(\lambda_n)$, the series, whose general term is $\lambda_n a_n$, is commutatively convergent in $E$, and that the linear mapping $(\xi_n) \mapsto \sum \xi_n a_n$ of the Banach space $\mathscr{B}(\mathbf{N})$ in $E$ is injective and continuous. c) Deduce from b) that when $E$ is an infinite dimensional Fréchet space then every basis of $E$ on $\mathbf{R}$ has cardinal at least equal to $2^{Card(\mathbf{N})}$ (cf. I, p. 22, exerc. 5). If there exists an enumerable set that is dense in $E$, then every basis of $E$ has the cardinal of the continuum.