I am reading a paper which currently gives me some confusion. The setup is like this (I do not need a prove for it. I am mostly curious if my intuition is right.)
I am looking at a set where is the set of all positive integers and the super index refers to an infinite dimension.
The vectors of this set are counting measures for another set ( is again the set of all integers.) A counting measure means: If some 's row element was with this would express that with this vecor 3 "entities" were counted (something of the model that is described in the paper) that have a value of . If it must be that for all vectors since no elements of this type can even exist.
The author now states that and must be chosen such that is compact. I am wondering: Isn't already compact? I darkly remember some theorem. In any case: is compact, or am I wrong? If so: Must be finite such that is automatically compact?
Help is very appreciated!