Let $X$ be a Banach space of infinite dimension. And let $K\subset X$ be a compact subset of $X$. Can we conclude something about the interior of $K$? Is it true that it's empty? I don't know how to attack this problem. I have not even examples of compacts on the infinite dimensional case.
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1Where does $X$ come into play? – Daniel Donnelly Oct 09 '13 at 03:08
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X has infinite dimension, I want to know if there exist a compact subset of X that has nonempty interior – Shanks Oct 09 '13 at 03:09
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The interior of a compact set in an infinite dimensional space is necessarily empty.
If not, then there would be an open set with compact closure. You could translate that set to get a neighbourhood of any other point with compact closure. Then the space is locally compact.
However, Any locally compact topological vector space must be finite dimensional. This should be proved in pretty much any reasonable book on Functional Analysis.
Prahlad Vaidyanathan
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