# Tensor product

Why these two tensor products are isomorphic?

$$Hom_{K}(V,K) \otimes Hom_{K}(V,K)$$ and $$Hom_{K}(V \otimes V,K)$$

where K is a field and V is a vector space over K.

quasar987
Homework Helper
Gold Member
This follows from the following elementary facts about tensors product that you should try to prove as easy exercices.

For V,W vectors spaces over K, and o the tensor product,

(1) Hom(V,W) is naturally isomorphic to V* o W
(2) V o K is naturally isomorphic to V
(3) (V o W)* is naturally isomorphic to V* o W*