$C(S \times T)$ is isomorphic to $C(S) \otimes C(T).$

Question

Let $S$ and $T$ be two arbitrary sets and consider the vector spaces $C(S)$ and $C(T)$ generated respectively by S and T. Show that $C(S \times T)$ is isomorphic to $C(S) \otimes C(T).$

I am starting to read Werner Greub's Multilinear Algebra and I come across this exercise, I have tried to find a bilinear mapping that associates these two vector spaces with me, can you help me?

Answer 1

By the universal property of the free space, there is a unique linear map $\varphi:C(S\times T)\to C(S)\otimes C(T)$ with $\varphi(s,t)=s\otimes t$ for $s\in S$ and $t\in T$.

Observe that $\varphi$ is injective since $$0=\varphi\bigl(\,\sum_i\lambda_i(s_i,t_i)\bigr)=\sum_i\lambda_i\,s_i\otimes t_i$$ implies $\lambda_i=0$ by linear independence of the $s_i$ in $S$ and the $t_i$ in $T$ (1.5.1). Also $\varphi$ is surjective since $$\bigl(\,\sum_i\lambda_i s_i\bigr)\otimes\bigl(\,\sum_j\mu_j t_j\bigr)=\sum_{i,j}\lambda_i\mu_j\,s_i\otimes t_j=\sum_{i,j}\lambda_i\mu_j\,\varphi(s_i,t_j)$$ and the elements on the left generate $C(S)\otimes C(T)$.