Exercise 6, § 4

Let E be the locally convex metrizable space $\ell^1$($\mathbf{N}$) (I, p. 4) endowed with the topology induced by that of the product space $\mathbf{R}$^$\mathbf{N}$; its dual E' can be identified with $\mathbf{R}$^{($\mathbf{N}$)} and the topology τ(E', E) is the topology induced on E' by the norm topology of c_0($\mathbf{N}$) (IV, p. 47, exerc. 1). Show that if u is a surjective continuous linear mapping from E onto a Hausdorff barrelled space F, then F is necessarily finite dimensional. (Observe that 'u is an isomorphism from F', endowed with σ(F', F), onto a subspace of E', endowed with σ(E', E); from the fact that F is barrelled, conclude that 'u(F') endowed with the topology induced by τ(E', E), is a Banach space, and use exerc. 24 of II, p. 80.)