I've been reading about functional analysis and topology and I came across this sentence:
subset of a topological vector space is compact with respect to the weak topology
I have been trying to search the web for an explanation to understand what this means, and I also tried the chat rooms without finding a satisfactory answer. English is not my native language so that might be one of the reasons I'm not getting the sentence. I do understand the concepts of topological space, compactness and weak topology.
My question is: What does this sentence mean, explicitly?
If you know weak topology the statement simply means any open cover of the subset by open sets in the weak topology has a finite subcover. For example, the closed unit ball in an infinite dimensional Hilbert space is not compact in the norm topology but it is compact in the weak topology.