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In reply to "Not complete", posted by Brian Green on April 25, 2010:
>I need to show that if I have a vector space X, that has countably infinite dimension. I need to show that there does not exist any norm that will make X complete.
>
>My proof so far, I have started with a norm ||.||_1 that makes X complete. Then I choose a finite dimensional subspace of X, say E, then I know E is closed since every finite dimensional subspace of X is closed.
>Here, I am trying to make the Baire Category Thm fail, which would then say that X cannot be complete, so there must not exist such a norm. I am not sure of why my closed set will make it fail? I know the sets need to be open and dense, but I can't fill in the details in the middle.
"me" helped with the Baire category argument. That's enough to see that an infinite dimensional Banach space cannot have countable linear dimension. More is true: an infinite dimensional Banach space has linear dimension at least 2^{\aleph_0} (independent of CH). For the stronger result, see:
The Hamel Dimension of any Infinite Dimensional Separable Banach Space is c
Author(s): H. Elton Lacey
Source: The American Mathematical Monthly, Vol. 80, No. 3 (Mar., 1973), p. 298
Published by: Mathematical Association of America
Stable URL: http://www.jstor.org/stable/2318458
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