Real Vector Space: Is Addition & Scalar Multiplication Smooth?

[Geometry_dude]

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.

 

[dextercioby]

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.

 

[Geometry_dude]

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?

 

[dextercioby]

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.

 

[Geometry_dude]

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