I have a couple of questions about tensor products:
Why is Hom(V,W) the same thing as V∗⊗W?
Why is an element of V∗⊗m⊗V⊗n the same thing as a multilinear map Vm→V⊗n?
What is the general formulation of this principle?
The result is generally wrong for infinite-dimensional spaces: see this question.
For finite dimensional space V, let’s build an isomorphism f:V∗⊗W→hom(V,W) by defining
This clearly defines a linear map V∗⊗W→hom(V,W) (it’s bilinear in V∗×W). Reciprocally, take a basis (ei) of V, then define g:hom(V,W)→V∗⊗W by:
Where (e∗i) is the dual basis to (ei) (I will use a few of its properties in red below). This is well-defined because V is finite-dimensional (the sum is finite). Let’s check that f and g are inverse to each other:
For u:V→W, f(g(u))(v)=∑e∗i(v)u(ei)=u(∑e∗i(v)ei)=u(v) and so f(g(u))=u.
For ϕ⊗w∈V∗⊗W, g(f(ϕ⊗w))=∑e∗i⊗f(ϕ⊗w)(ei)=∑e∗i⊗ϕ(ei)w=∑ϕ(ei)e∗i⊗w=ϕ⊗w
And so f and g are isomorphisms, inverse to each other.
It is known that for finite dimensional V, then (V∗)⊗m=(V⊗m)∗. Then an element of V∗⊗m⊗V⊗n is an element of (V⊗m)∗⊗V⊗n=hom(V⊗m,V⊗n). So by definition / universal property of the tensor product, it’s a multilinear map Vm→V⊗n.