# Adv. linear algebra, tensor product, dual space

1. Nov 18, 2008

### jacobrhcp

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*$$\otimes$$W->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*$$\otimes$$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?