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Adv. linear algebra, tensor product, dual space

  1. Nov 18, 2008 #1
    1. The problem statement, all variables and given/known data

    Prove that V* [tex]\otimes[/tex] W is isomorphic to Hom(V,W) in the case that one of V and W is finite-dimensional.

    3. The attempt at a solution

    A pair (l,w) in V*xW defines a map V->W, v->l(v)w. This map is bilinear.

    Because it's bilinear, it defines a bilinear map V* [tex]\otimes[/tex] W -> Hom(V,W)

    Now if both were finite-dimensional, we could just pick a basis for both V and W, so (v,w) is the same as

    ([tex]\sum v^i e_i, \sum w^j f_j [/tex], and have the map V*[tex]\otimes[/tex]W->Hom(V,W) send [tex]x^i \otimes f_j[/tex] to [tex](u_i)^j:V->W[/tex] that takes e_i to f_j and every other basis vector e_i' to 0. now {u} makes up a basis for Hom(V,W) and because the map was linear I am done.

    but in the partially infinite dimensional case I cannot pick a basis.

    Suppose for instance W is finite. My best guess is to write an element of V*[tex]\otimes[/tex]W explicitly, find a better explicit mapping and find an explicit inverse. Then I am done. But how to do this?

    My guess: 'physics slang bracket notation' suggest a mapping like |w><v| -> l(v)w... but I got nothing explicit. Anyone wanna help me a bit?
  2. jcsd
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