Hello :) i think the tensor product is very difficulty, and i have some questions (let $U$ be any $k$-vectorspace and $k$ a field)
- How to construct a natural bijection (thus without choosing a basis) between bilinear maps $V\times W\rightarrow U$ and linear maps $V\otimes W\rightarrow U$?
- How to construct a natural isomorphism $V^*\otimes W\rightarrow Hom(V,W)$ (with V finite dimensional and $V^*$ the dual space of $V$.
I don't see how to produce such bijections and isomorphims. I think thats not difficult but i can't imagine such a mapping. Someone who can help me? Thanks.