Can anyone explain the relationship between uniform boundedness and equicontinuity with the compact operator?

Answer 1

Let F be a family of real-valued functions defined on a compact metric space X. Then F is relatively compact if and only if F is uniformly bounded and equicontinuous. This means that the closure of F is compact if and only if F is uniformly bounded and equicontinuous.

If you specialize to the case when X = [a,b] is a closed interval on the real line, then a family of functions F is just a family of functions defined on this interval, that is, each element of F is a function f from [a,b] to R. The theorem above says that the closure of F is compact if and only if F is uniformly bounded and equicontinuous. This is the so-called Arzela-Ascoli Theorem.