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redrzewski
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This is from Rudin, Functional Analysis 2.1. Not homework.
If X is an infinite-dimensional topological vector space which is the union of countably many finite-dimensional subspaces, prove X is first category in itself.
What about this example? Take R^n (standard n-dimensional space of reals) as each of the finite-dimensional subspaces. Then the union as n goes from 1 to infinity will be R^w.
R^w is infinite-dimensional, and it will contain closed sets that have non-empty interior. R^w seems like it will satify the axioms of the topological vector space. Hence this would contradict the problem.
What am I missing?
thanks
If X is an infinite-dimensional topological vector space which is the union of countably many finite-dimensional subspaces, prove X is first category in itself.
What about this example? Take R^n (standard n-dimensional space of reals) as each of the finite-dimensional subspaces. Then the union as n goes from 1 to infinity will be R^w.
R^w is infinite-dimensional, and it will contain closed sets that have non-empty interior. R^w seems like it will satify the axioms of the topological vector space. Hence this would contradict the problem.
What am I missing?
thanks