Exercise 23, § 4

Let X be a real infinite dimensional Banach space. a) Suppose that for every λ > 0, there exists a finite dimensional subspace E_λ of X such that there exists no continuous projector P on X with image E_λ and such that \|P\| ≤ λ. Show that every closed subspace Y of X with finite codimension has the same property as X. b) By induction on n, show that there exists a decreasing sequence (X_n) of closed subspaces of X, of finite codimension, and for every n, a finite dimensional subspace E_n of X such that : 1° the sum E_1 + E_2 + ⋯ + E_n is direct, and there exists a continuous projector P_n on X_n, with norm ≤ 2 whose image is E_1 + E_2 + ⋯ + E_n; 2° the space E_n is contained in (I − P_{n−1})(X_{n−1}); 3° there exists no continuous projector on X with image E_n and with norm ≤ n + 2. c) Let Z be the closed subspace of X generated by the union of the E_n. Show that there does not exist a topological complement of Z in X (observe that if Q were a continuous projector on X, with image Z, then (I − P_{n−1})P_nQ would be a continuous projector on X with image E_n).