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Smooth Vector Space

  1. Sep 24, 2014 #1
    Let ##V## be a real vector space and assume that ##V## (together with a topology and smooth structure) is also a smooth manifold of dimension ##n## with ##0 < n < \infty##, not necessarily diffeomorphic or even homeomorphic to ##\mathbb R^n##.

    Here's my question: Does this imply that addition and scalar multiplication is smooth?

    I tried to find a counterexample and thought about exotic ##\mathbb R^4##, but my knowledge about that is quite limited.
     
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  3. Sep 24, 2014 #2

    dextercioby

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    Any real vector space of dimension n<infinity is necessarily isomorphic to R^n, as vector spaces. I think the manifold you're trying to imagine is necessarily diffeomorphic to R^n.
     
  4. Sep 24, 2014 #3
    That there is a vector space isomorphism to ##\mathbb R^n## is not disputed, yet this does not necessarily mean that it is a homeomorphism or diffeomorphism when we consider ##\mathbb R^n## with the standard smooth structure and topology.

    EDIT: Maybe group theory holds the answer?
     
  5. Sep 24, 2014 #4

    dextercioby

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    Alright then, let's assume I am wrong: offer me an example of a vector space endowed with a topology (what type ?) and a smooth structure that is not diffeomorphic to R^n with the usual topology and differential structure.
     
  6. Sep 24, 2014 #5
    You know very well that this is not the way mathematics works.

    I do know one though, the one in the opening post:
    http://en.wikipedia.org/wiki/Exotic_R4
     
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