I'm trying to solve exercise 2 paragraph 25 from "Finite dimensional vector spaces" by Paul R. Halmos
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 $\le m - 1$ for each fixed $s$ and $\le n - 1$ for each fixed $t$. Prove that there exists an isomorphism between $P_n \otimes P_m$ and $P_{n,m}$ such that the element $z$ of $P_{n,m}$ that corresponds to $x \otimes y$ ($x \in P_n$ and $y \in P_m$) is given by $z(s, t) = x(s)y(t)$
Does this mean that any polynomial of two variables can be represented as a product of two polynomials of one variable?
It means that any polynomial $z(s,t)$ can be represented as a sum of polynomials, each of the form $x(s)y(t)$.