Exercise 7, § 4

a) Let E be a Banach space, and (x_α)_{α∈A} an everywhere dense set in the unit sphere of E. Let u be the linear mapping from the space $\ell^1$(A)(I, p. 4) into E defined by u(t) = $\sum_{\alpha\in A} t(\alpha)x_\alpha$ for all $t = (t(\alpha))_{\alpha\in A}$ belonging to $\ell^1$(A). Show that u is a strict morphism from $\ell^1$(A) onto E and consequently that E is isomorphic to a quotient space of $\ell^1$(A).

b) From a) deduce an example of a closed subspace of $\ell^1(\mathbf{N})$ which has no topological complement in $\ell^1(\mathbf{N})$ (for E take $c_0(\mathbf{N})$ and use IV, p. 54, exerc. 15, $b$) and p. 55, exerc. 18).