Compactness in function spaces

摘要

Let X be a locally compact topological space, (Y, d) be a boundedly compact metric space and LB(X, Y) be the space of all locally bounded functions from X to Y. We characterize compact sets in LB(X, Y) equipped with the topology of uniform convergence on compacta. From our results we obtain the following interesting facts for X compact:If {f(n) : n is an element of N} is a sequence of uniformly bounded finitely equicontinuous functions of Baire class alpha from X to R, then there is a subsequence {f(nk) : k is an element of N} uniformly convergent to a Baire class a function from X to R;If {f(n) : n is an element of N} is a sequence of uniformly bounded finitely equicontinuous lower (upper) semicontinuous functions from X to then there is a subsequence {f(nk) : k is an element of N} uniformly convergent to a lower (upper) semicontinuous function from X to R;If {f(n) : n is an element of N} is a sequence of uniformly bounded finitely equicontinuous quasicontinuous functions from X to Y, then there is a subsequence {f(nk) : k is an element of N} uniformly convergent to a quasicontinuous function from X to Y. (C) 2019 Elsevier B.V. All rights reserved.