Uniformity and Metrization

Abstract

In principle, weak convergence is the pointwise convergence of “operators” X _α or L _α on the space C _b (D). However, there is automatically uniform convergence over certain subsets. These subsets can be fairly big: equicontinuity and boundedness suffice. On the other hand, there also exist small (countable) subsets such that pointwise convergence on such a subset is automatically uniform, and equivalent to pointwise convergence on the whole of C _b (D), i.e. weak convergence. For separable D, it is even possible to pick such a countable subset that works for every X _α , at the same time.