THE UNIFORM FORWARD SETS OF $C(X)$ AND UNIFORM CONVERGENCE OF OPERATOR SERIES

Abstract

In this paper, we introduce the uniform forward sets of $C(X)$ and show that each totally bounded set of $C(X)$ is uniform forward set, moreover, we prove that the uniform forward sets of $C(X)$ are just the largest subset family of $C(X)$ on which each $C(X)$-evaluation convergent operator series is uniformly convergent.