Paper detail

Banach spaces with no proximinal subspaces of codimension 2

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&#39;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&#39;,f&#39;) in X x X* with ||x&#39;||= ||f&#39;||=1, ||x-x&#39;||, ||f-f&#39;||< h and f&#39;(x&#39;)=1. This means that there are always &#34;proximinal&#34; hyperplanes H in X (a nonempty subset E of a metric space is said to be &#34;proximinal&#34; 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 &#34;The Theory of Best approximation and Functional Analysis&#34; 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.