Adv. linear algebra, tensor product, dual space

[jacobrhcp]

Homework Statement

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

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 want to help me a bit?

 

[mighty2000]

Thank you for your post. I understand your confusion regarding the partially infinite dimensional case. However, I believe that with a little bit of clarification, you can still prove the isomorphism between V* \otimes W and Hom(V,W).

Firstly, let's define the elements of V* \otimes W in terms of the basis vectors. We can write an element (l,w) as a linear combination of basis vectors as (l,w) = \sum_i l_i \otimes w_i. Similarly, an element of Hom(V,W) can be written as a linear combination of basis elements as \sum_{i,j} a_{ij}u_i \otimes f_j, where a_{ij} are scalars and {u_i} and {f_j} are basis elements for V and W respectively.

Now, let's define the map \phi: V* \otimes W \to Hom(V,W) as follows:

\phi(\sum_i l_i \otimes w_i) = \sum_{i,j} l_i(u_j)f_i \otimes w_i.

This map is well-defined and linear. To show that it is an isomorphism, we need to show that it is one-to-one and onto.

One-to-one: Suppose \phi(\sum_i l_i \otimes w_i) = \sum_{i,j} l_i(u_j)f_i \otimes w_i = 0. This implies that for every j, \sum_i l_i(u_j)f_i = 0. Since {u_i} is a basis for V, this means that l_i = 0 for all i. Hence, \sum_i l_i \otimes w_i = 0, showing that \phi is one-to-one.

Onto: Let \sum_{i,j} a_{ij}u_i \otimes f_j be an element of Hom(V,W). We can write this as \sum_i (\sum_j a_{ij}f_j) \otimes u_i. Therefore, \phi(\sum_i (\sum_j a_{ij}f_j) \otimes u_i) = \sum_{i,j} a_{ij}u_i \otimes f_j, showing that \phi is onto.

Hence, \phi is an isomorphism, proving that V* \otimes W is isomorphic to Hom(V,W) in