- #1
Fractal20
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Homework Statement
So I am trying to understand this proof and at one point they state that an arbitrary compact subset of a Banach space, or a completely metrizable space is the subset of a finite set and an arbitrary convex neighborhood of 0. I've been looking around and can't find anything to support this. Where does this come from?
Homework Equations
The Attempt at a Solution
I keep thinking that maybe it can be approached by the set being totally bounded since it is compact. So it seems for a given ε can cover the subset in open balls of radius ε. Then the centers of these balls is a finite set, then somehow choosing a corresponding convex set might let you represent any of the elements of the original compact set as the sum of elements of these new sets. But I don't know, I just don't have any basis from which to approach this.