jacobrhcp
Nov18-08, 05:07 PM
1. The problem statement, all variables and given/known data
Prove that V* \otimes 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* \otimes 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
(\sum v^i e_i, \sum w^j f_j , and have the map V*\otimesW->Hom(V,W) send x^i \otimes f_j to (u_i)^j:V->W 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*\otimesW 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?
Prove that V* \otimes 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* \otimes 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
(\sum v^i e_i, \sum w^j f_j , and have the map V*\otimesW->Hom(V,W) send x^i \otimes f_j to (u_i)^j:V->W 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*\otimesW 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?