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linear algebra - What is the standard proof that dim(k^N is uncountable? - Mathematics Stack Exchange: asked Jul 29 '12 at 13:46 Chindea Filip

What is the standard proof that dim(kN)is uncountable?
This is my (silly) proof to a claim on top of p. 54 of Rotman's "Homological algebra".
略


1 Answer answered Jul 29 '12 at 14:29 Asaf Karagila
One liner argument which uses a much more difficult theorems (swatting gnats with cluster bombs kind of proof):
kN is the algebraic dual of the polynomials in one variable, k[x] which has a countable dimension. If kN had a countable basis then k[x] would be isomorphic to its dual, and since this cannot be we conclude that kN has a basis of uncountable size.

The arguments given in Arturo's answer show that the above is indeed a proof (in particular Lemma 2 with κ=?0 ).