Vol. 25, No. 2, 1968

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ISSN: 0030-8730
Extreme points and dimension theory

Newton Tenney Peck

Vol. 25 (1968), No. 2, 341–351
Abstract

The purpose of this paper is to characterize the topological dimension of a compact metric space X in terms of the extremal structure of the unit ball of the spaces C(X,Rn), where Rn denotes Euclidean n-space with the usual Euclidean norm and C(X,Rn) denotes the space of continuous maps of X into Rn, normed by the sup norm. The main results are that if n 2, the unit ball of C(X,Rn) is always the closed convex hull of its extreme points, and that if the unit ball of C(X,Rn) is actually equal to the convex hull of its extreme points, then the dimension of X is less than n. If n is even, the converse of the second assertion above is shown to be true, and with additional assumptions on X, the converse of the second assertion holds whether n is even or odd.

In the last half of the paper, the corresponding questions for the spaces C(X,N) are studied, where N is an infinite-dimensional strictly convex normed space and C(X,N) is the space of continuous maps of X into N, again with the sup norm. Here it is established that the unit ball of C(X,N) is always the convex hull of its extreme points.

Mathematical Subject Classification
Primary: 54.70
Secondary: 46.00
Milestones
Received: 7 June 1966
Revised: 10 March 1967
Published: 1 May 1968
Authors
Newton Tenney Peck