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It shows that V is the dual space of V*?

  1. Jul 22, 2007 #1
    Not my homework
    It's in the textbook - lectures of diff geo by s.s.chern
    Just put them down in a clearer way
    Could anybody explain the questions in the pic ?
    http://x7d.xanga.com/be6d931344030137094065/z100635368.jpg
     
  2. jcsd
  3. Jul 22, 2007 #2

    matt grime

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    This is only true if V is finite dimensional. Hint just write down bases.
     
  4. Jul 22, 2007 #3
    actually i dont understand the last step
    Phi is taken out of the sum because it is linear in the 2nd variable ?
    And "V is the dual space of V*"looks very confusing
    The dual space of V is the set linear functions on V
    Now the dual space of V* is the set linear functions on V*?
    how can i see that ?
     
  5. Jul 22, 2007 #4
    i just got another idea
    since <v,v*> just depends on v*
    so we just treat it as a function of v* and then we have the last step ?
     
  6. Jul 23, 2007 #5

    matt grime

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    If V is a finite dimensional vector space, then V the only invariant of V is its dimension. ALL vector spaces of a given dimension (over the same field) are isomorphic. Since V is clearly isomorphic to V* by picking a basis and corresponding dual basis, this shows that (V*)* must be isomorphic to V since they have the same basis. (This fails in infinite dimensions.)

    What I presume you're looking at is a 'nice' bijection that sends v in V to the function on V* that sends f to f(v). Notice how an element of v can be sent to a function, call it e_v, and think of it as evaluation at v. This is a linear functional on the space of linear functionals:

    e_v(f)=f(v).

    This defines a map from V to the space of linear functionals on V*

    v--->e_v

    it is straight foward to show this is an isomorphism.
     
  7. Jul 31, 2007 #6

    mathwonk

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    if f(x) is a number, then f sends x to a number, and x sends f to a number. so f is a function of x, and x is a function of f. so points of the domain are functions on functions.
     
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