Exercise 9, § 4

Let (r_n)_{n ≥ 1} be the sequence of distinct rational numbers belonging to the interval I = [0, 1] of R, arranged in a certain order. One defines by recurrence a sequence of closed intervals I_n ⊂ I as follows: I_n has as midpoint the point r_{k_n} of smallest index not contained in the union of the intervals I_p of index p < n; it has length ≤ 1/4^n and meets none of the I_p of index p < n; the I_n therefore form a sequence of pairwise disjoint closed intervals. In the product space I × R, one defines a continuous numerical function u as follows: for every integer n ≥ 1, the function x ↦ u(x, n) is equal to 1 at an interior point of I_n, to 0 at every exterior point of I_n and takes its values in {0, 1}; moreover, for each x ∈ I, the function y ↦ u(x, y) is affine linear in each of the intervals [n, n + 1]. Now prove that u cannot be approximated uniformly in I × R by a linear combination of functions of the form v(x)w(y), where v is continuous in I and w continuous and bounded in R. (Consider, in the space $C^b(\mathbf{R}; \mathbf{R})$ of continuous and bounded functions in R, the set of partial functions y ↦ u(x, y) when x runs through I; show that there exists an infinite sequence (u_n) of these functions such that \|u_n\| = 1 and \|u_m - u_n\| = 1 for m ≠ n. Deduce that there could not exist a vector subspace E of $C^b(\mathbf{R}; \mathbf{R})$, of finite dimension, such that each of the u_n is at a distance ≤ 1/4 from this subspace; otherwise, there would exist a sequence (v_n) of points of E such that \|v_n\| = 2 and \|v_n - v_m\| ≥ 1/2 for m ≠ n, contrary to the fact that every finite dimension subspace of $C^b(\mathbf{R}; \mathbf{R})$ is locally compact).*