Exercise 15, § 4

a) Let E be an infinite dimensional normed space (real or complex). Show that there exists a sequence $(x_n)$ in E such that, for every bounded sequence $(\lambda_n)$ of scalars, there exists a continuous linear form $x'$ on E such that $\langle x_n, x' \rangle = \lambda_n$ for all $n$. (Construct a sequence $(x'_n)$ of points in the dual $E'$ of E and a sequence $(x_n)$ of points in E such that $\langle x_i, x'_j \rangle = \delta_{ij}$ and $\|x'_n\| \leq 2^{-n}$.) b) For a locally convex metrizable space E to have the property stated in a), it is necessary and sufficient that the completion of E is not a space of minimal type (II, p. 85, exerc. 13).