Homeomorphism between a 1-dim vector space and R

[jem05]

im trying to get a homeomorphism between a 1-dim vector space and R, but independent of the basis.
Any ideas?

 

[mrbohn1]

Do you mean homomorphism? Homeomorphisms are maps between topological spaces.

 

[jem05]

no i meant homeomorphism,
im trying to get a chart (manifolds- C^(/infinity) structure) from V to R
we can though define the norm right and induce a topology on our vector space,
but i still can't see the homeomorphism,
ofcourse the hard thing is that i want it to be independent of the basis, or else, it would've been trivial.

 

[Office_Shredder]

I would be pretty surprised if you could find one that's basis free.

You can't even define the norm without a basis unless you have a basis free definition for a linear isomorphism between V and R.

 

[Hurkyl]

Can you give a precise statement of what you mean by "independent of basis"?

 

[jem05]

yeah sure, i think they mean for whatever choice of bases, we can get the same result.

 

[Hurkyl]

Then pick a particular basis, and use that to define an isomorphism of vector spaces.

Once you have done that, you have defined a function. The value of the function depends only on the point chosen and nothing else.

(Of course, the obvious method for computing values of this function would probably make use of the basis you originally chose)