Compact Sets of Functions and Function Rings

A widely used theorem of analysis asserts that a uniformly bounded, equicontinuous family of functions has a compact closure in the space of continuous functions. This lemma, variously attributed to Arzela, Escoli, Montel, Vitali, and so on, is of importance in the theory of integral equations, conformal mapping, calculus of variations, and so on. In recent years the lemma has been generalized by S. B. Myers [l].1 A part of his results may be formulated as follows; If a topological space X is either (a) locally compact, (b) satisfies the first axiom of countability, and if Y is a metric space, then a family F of continuous functions from X to Y is compact (in a suitable topology) if and only if (1) F(x) = (J/^Ff(x) is compact for all xG.X, (2) Pis closed, (3) Pis equicontinuous. The main purpose of §1 of this paper is to characterize compact sets of functions when Y is any regular topological space. The problem is therefore to find a condition to replace equicontinuity, which no longer makes sense. We obtain such a characterization which holds for an easily described class of spaces X which includes both locally compact and first countable. In §2 these results are applied to obtain a sort of duality theorem for the ring of real-valued continuous functions, R(X), on a space X. Namely, it is shown that, under quite general conditions, the space X is homeomorphic with the space H(X) of continuous homomorphisms from the ring R(X) onto the real numbers, where R(X) and H(X) are given the "compact-open" topology.