Originally Posted by

**cduston** Hey all,

I would like to prove the following:

$\displaystyle C(X \times Y)=C(X)\otimes C(Y),$

where $\displaystyle C(X)$ are continuous, complex-valued functions on Hausdorff space $\displaystyle X$. I don't want to specify them as bounded, but I think my solution requires that.

The first question is what '=' means here, so it at least means isomorphic, and probably isometric too. So if someone wants to just comment about that part it would be appreciated (like are isomorphic Banach spaces always isometric or something).

So my thought for the isomorphism is something like the map taking

$\displaystyle f\otimes g \to F\in C(X\otimes Y)\mbox{ where } F(x,y)=f(x)g(y).$

Then I think showing something like the subalgebra $\displaystyle f\otimes g\subset C(X)\otimes C(Y)$ separates $\displaystyle C(X\otimes Y)$, and then invoke the Stone-Weierstrass theorem.

Does this make sense to anyone? And is it common convention to assume "isomorphic Banach algebras" are "isomorphic, isometric Banach algebras"?