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Mar26-11, 03:28 PM
Sci Advisor
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mathwonk's Avatar
P: 9,488
This may be stupid but it seems obvious that a Q vector space with a countable basis is countable. Thus R is an uncountable dimensional Q vector space. The same argument should show the dimension of R equals card(R). Since C has the same cardinality as R, it cannot have larger dimension.

I think the missing details are somewhere around the words "obvious that..."