# Stone-Weierstrass theorem problem

1. Dec 6, 2004

### cogito²

I'm working on a problem that has to do with the Stone-Weierstrass theorem. This is the problem:

The way that I've been trying to do it is to produce an algebra of continuous functions that separates points and contains constant functions. If I define $A$ to be the set of all \sum_{i=1}^n g_ih_i where $g_1,\ldots,g_n$ are continuous on $X$ and $$h_1,\ldots,h_n$$ are continuous on $Y$, it is easy to show that constant multiples of functions in $A$ are in $A$, $A$ is closed under multiplication, $A$ separates points, and $A$ contains the constant functions. What I am having trouble showing is that $A$ is closed under addition (ie. that $A$ actually is an algebra). Is this true? If it is not then does anybody know of a way to come up with an algebra for this problem so that I could apply Stone-Weierstrass? Any help would be greatly appreciated.

2. Dec 6, 2004

### cogito²

Okay now I feel stupid. Now thinking about it sums are included basically by definition.