  The classical theorem of Bishop-Phelps asserts that, for a Banach space X,
the norm-achieving functionals in X* are dense in X*. Bela Bollobas's extension
of the theorem gives a quantitative description of just how dense the
norm-achieving functionals have to be: if (x,f) is in X x X* with ||x||=||f||=1
and |1-f(x)|< h^2/4 then there are (x',f') in X x X* with ||x'||= ||f'||=1,
||x-x'||, ||f-f'||< h and f'(x')=1.
  This means that there are always "proximinal" hyperplanes H in X (a nonempty
subset E of a metric space is said to be "proximinal" if, for x not in E, the
distance d(x,E) is always achieved - there is always an e in E with
d(x,E)=d(x,e)); for if H= ker f (f in X*) then it is easy to see that H is
proximinal if and only if f is norm-achieving. Indeed the set of proximinal
hyperplanes H is, in the appropriate sense, dense in the set of all closed
hyperplanes H in 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 show that there is a
Banach space X such that X has no proximinal subspace of finite codimension
n>1. So we have a converse to Bishop-Phelps-Bollobas: a dense set of proximinal
hyperplanes can always be found, but proximinal subspaces of larger, finite
codimension need not be.
