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>>628 > >>627 再投稿 > > http://math.stackexchange.com/questions/176475/what-is-the-standard-proof-that-dimk-mathbb-n-is-uncountable > 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 ).
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