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Homeomorphism between a 1-dim vector space and R

  1. Mar 23, 2010 #1
    im trying to get a homeomorphism between a 1-dim vector space and R, but independent of the basis.
    Any ideas?
     
  2. jcsd
  3. Mar 23, 2010 #2
    Re: homeomorphisms

    Do you mean homomorphism? Homeomorphisms are maps between topological spaces.
     
  4. Mar 23, 2010 #3
    Re: homeomorphisms

    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 cant 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.
     
  5. Mar 24, 2010 #4

    Office_Shredder

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    Re: homeomorphisms

    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.
     
  6. Mar 24, 2010 #5

    Hurkyl

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    Re: homeomorphisms

    Can you give a precise statement of what you mean by "independent of basis"?
     
  7. Mar 25, 2010 #6
    Re: homeomorphisms

    yeah sure, i think they mean for whatever choice of bases, we can get the same result.
     
  8. Mar 25, 2010 #7

    Hurkyl

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    Re: homeomorphisms

    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)
     
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