What is a tensor product? What is the universal property? What do the tensors look like in the case of vector spaces?
Solution
1. \(V\otimes W\) is the quotient of free vector space on \(V\times W\) by bilinear relations.
construction.
2. Universal property: bilinear \(B:V\times W\to U\) factors uniquely through \(V\otimes W\) via \(v\otimes w\mapsto B(v,w)\).
universal property.
3. For vector spaces with bases \(\{e_i\},\{f_j\}\), \(V\otimes W\) has basis \(\{e_i\otimes f_j\}\), and tensors are finite sums \(\sum c_{ij}e_i\otimes f_j\).
basis.
4. Q.E.D.
2 and 3.