Is $\mathbb{R}^2 = \mathbb{C}?$ are same set, metric space, vectorial space, field, etc? Thanks in advance.
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"vectorial" is the best alternative spellings I have seen :-) – Dontknowanything Sep 16 '15 at 18:50
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@Learner What's the standard spelling for "vectorial"? :) – Austin Mohr Sep 16 '15 at 19:03
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@AustinMohr Isn't Vector Space? – Dontknowanything Sep 17 '15 at 15:59
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$\mathbb{R}^2 \cong \mathbb{C}$ as a real inner-product space, with all that entails regarding metrics and topology.
On the other hand, $\mathbb{R}^2 \cong \mathbb{C}$ as a field if we introduce the product $$ (a,b) \cdot (c,d) = (ac-bd,ad+bc), $$ which one can show does turn $\mathbb{R}^2$ into a field.
Chappers
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It is certainly not the same set, essentially (but not formally) the same metric space (they are homeomorphic), essentially (but not formally) the same vector space (they are linearly isomorphic) and as to field structure, the standard algebra on $\mathbb{R}^2$ is none.
Damian Reding
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