Let $K$ and $L$ be compact spaces. Consider the Banach algebra $V(K,L)=C(K)\mathbin{\hat\otimes} C(L)$ , which is the completion of the $C(K)\otimes C(L)$ with respect to the projective tensor norm. It is known that $V(K,L)$ is a subalgebra of $C(K\times L)$ , not isometrically. I am trying to establish this result . Heres what I have figured out. Due to the universal property of tensor product , there is an algebra homomorphism $\theta : C(K)\otimes C(L)\to C(K\times L)$ such that $$\theta(\sum_{i=1}^nf_i\otimes g_i)(x,y)=\sum_{i=1}^nf_i(x)g_i(y)$$
I want to show that this map is injective. Once this is done then we can extend it to its completion.
Suppose $h\in C(K)\otimes C(L)$ and $\theta(h)=0$. Write $h=\sum_{i=1}^nf_i\otimes g_i$, where the $g_i$ are linearly independent. But then this implies that for all $x\in K$, $$0=\theta(h)(x,\cdot)=\sum_{i=1}^nf_i(x)g_i(\cdot),$$ whence $f_i=0$ for all $i$, and thus $h=0$.