Banach spaces with no proximinal subspaces of codimension 2

Abstract

The classical theorem of Bishop—Phelps asserts that, for a Banach space X, the norm-achieving functionals in X* are dense in X*. Béla Bollobás’s extension of the theorem gives a quantitative description of just how dense the norm-achieving functionals have to be: if (x,φ) ∈ X × X* with ‖x‖ = ‖φ‖ = 1 and |1 - φ(x)| < ε²/4, then there are (x′, φ′) ∈ X × X* with ‖x′‖ = ‖φ′‖ = 1, ‖x - x′‖ < ε, ‖φ - φ′‖ < ε and φ′(x′) = 1.

This means that there are always “proximinal” hyperplanes H ⊂ X (a nonempty subset E of a metric space is said to be “proximinal” if, for x ∉ E, the distance d(x, E) is always achieved—there is always an e ∈ E with d(x,E) = d(x, e)); for if H = kerφ (φ ∈ X*), then it is easy to see that H is proximinal if and only if φ is norm-achieving. Indeed the set of proximinal hyperplanes H is, in the appropriate sense, dense in the set of all closed hyperplanes H ⊂ X.

Quite a long time ago [Problem 2.1 in his monograph The Theory of Best Approximation and Functional Analysis, Regional Conference Series in Applied Mathematics, SIAM, 1974], Ivan Singer asked if this result generalized to closed subspaces of finite codimension—if every Banach space has a proximinal subspace of codimension 2, for example. In this paper I will show that there is a Banach space X such that X has no proximinal subspace of finite codimension n ≥ 2. So we see that the Bishop–Phelps–Bollobás result is sharp: a dense set of proximinal hyperplanes can always be found, but proximinal subspaces of larger, finite codimension need not exist.