1. The problem statement, all variables and given/known data Let P(n,m) be the space of all polynomials z with complex coefficients, in two variables s and t, such that either z = 0 or else the degree of z(s, t) is <= m - 1 for each fixed s and <= n - 1 for each fixed t. Prove that there exists an isomorphism between Pn (x) Pm (tensor product of Pn and Pm) and P(n,m) such that the element z of P(n,m) that corresponds to a (x) b (tensor product of vectors, a in Pn, b in Pm) is given by z(s,t) = a(s)b(t). 3. The attempt at a solution Never mind the definition of tensor product, it seems to be that the conclusion can't be correct? because if there is such an isomorphism then every element z in P(n,m) can be written as z(s,t) = a(s)b(t) where a and b are polynomials, however z(s,t) = s + t clearly is an element P(n,m) but cannot be written in this form? There's probably something that i'm not getting so I would appreciate it if anyone can point it out for me. Thanks.