i want to understand a proof of $C(X)\otimes_{\text{min}}C(Y)\cong C(X\times Y)$ for compact Hausdorff spaces $X,Y$ via the Gelfand transform. Using the latter, it suffices to show that $\Omega(C(X)\otimes_{\text{min}}C(Y))$ and $C(X\times Y)$ are homeomorphic, where $\Omega$ denotes the Gelfand spectrum.
Now, we define $\gamma_{(x,y)}:C(X)\times C(Y)\to\mathbb{C},(f,g)\mapsto f(x)g(y)$. Since this is bilinear, we obtain a linear map $\widetilde{\gamma_{(x,y)}}:C(X)\otimes_{\text{alg}} C(Y)\to\mathbb{C}$. The following step is the one i don't understand: We want to prove that $\widetilde{\gamma_{(x,y)}}$ is bounded, so that it extends to a linear continous map defined on $C(X)\otimes_{min} C(Y)$. I can prove the boundedness for simple tensors, but not for sums of those. How can I do that? The remainder of the proof is clear to me.
I know that there are other proofs of the statement itself, but I'm particularly interested in how to obtain the boundedness as this is a general problem when extending maps from the algebraic to the minimal tensor product.