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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.
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