A remark on the intersection of shells

Abstract

There are many characterizations which distinguish finite-dimensional normed linear spaces from infinite-dimensional normed linear spaces. Perhaps the best known of these is the compactness of the unit ball. Recently, V. Klee [1] showed that in any infinite-dimensional normed linear space there exists a decreasing sequence of unbounded but linearly bounded closed convex sets whose intersection is empty. We will give here a somewhat similar condition which holds in all infinite-dimensional normed linear spaces but does not hold in any finite-dimensional space. We begin with the following terminology. U will denote the unit ball {x: ||x|| S 1 } and S the unit sphere {x: ||x|| = 1 }. A shell will be any set of the form {x: ri? IIx-xalI ? r2 for 0 1 such that (x+p U)Cr (-x+p U) CO, and S will be called finitely nonfiat if there exists a finite number of points {xi} '1 in S such that n(xj+pU) Co. A collection of sets has the finite intersection property if the intersection of the sets in any finite subcollection is not empty.