Compactness in function spaces

TL;DR

This paper characterizes compact sets in spaces of locally bounded functions with uniform convergence on compacta, providing new subsequence convergence results for various classes of functions on compact spaces.

Contribution

It offers a characterization of compact sets in function spaces with the topology of uniform convergence on compacta, leading to new subsequence convergence theorems for specific function classes.

Findings

Sequences of uniformly bounded finitely equicontinuous functions have uniformly convergent subsequences.

Sequences of semicontinuous functions with boundedness and equicontinuity have uniformly convergent subsequences.

Sequences of quasicontinuous functions with boundedness and equicontinuity have uniformly convergent subsequences.

Abstract

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: $\bullet$ If $(f_n)_n$ is a sequence of uniformly bounded finitely equicontinuous functions of Baire class $\alpha$ from $X$ to $\Bbb R$, then there is a uniformly convergent subsequence $(f_{n_k})_k$; $\bullet$ If $(f_n)_n$ is a sequence of uniformly bounded finitely equicontinuous lower (upper) semicontinuous functions from $X$ to $\Bbb R$, then there is a uniformly convergent subsequence $(f_{n_k})_k$; $\bullet$ If $(f_n)_n$ is a sequence of uniformly bounded finitely equicontinuous quasicontinuous functions from $X$ to $Y$, then there is a uniformly convergent subsequence $(f_{n_k})_k$.