Stone-Weierstrass theorem problem

[cogito²]

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

Let $X$ and $Y$ be compact spaces. Then for each continuous real-valued function $f$ on $$X \times Y$$ and each $$\epsilon > 0$$ there exist continuous real-valued functions $g_1,\ldots,g_n$ on $X$ and $$h_1,\ldots,h_n$$ on $Y$ such that for each $$(x,y) \in X \times Y, |f(x,y) - \sum_{i=1}^n g_i(x)h_i(y)| < \epsilon.$$

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.

 

[cogito²]

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

 

[wais]

The Stone-Weierstrass theorem is a powerful tool in analysis and it is great that you are working on a problem related to it. Your approach of constructing an algebra of continuous functions that separates points and contains constant functions is a good start. However, as you have mentioned, the key to applying the Stone-Weierstrass theorem is to show that your algebra is closed under addition.

To show this, you can use the fact that the product of two continuous functions is also continuous. So, if you have two functions in your algebra A, say f and g, then their product fg is also in A. Then, you can use the fact that the sum of two functions can be written as the difference of two products: (f+g) = (f+g) - (f-g). Since A is closed under multiplication and subtraction, it follows that A is also closed under addition.

If you are still having trouble showing that A is an algebra, you can try to construct a different algebra that satisfies the conditions of the Stone-Weierstrass theorem. For example, you can try to construct an algebra of trigonometric functions or polynomial functions that separates points and contains constant functions. It may also be helpful to look at examples of problems where the Stone-Weierstrass theorem has been applied to gain some insight into how to construct an appropriate algebra.

Overall, it is important to carefully consider the conditions of the Stone-Weierstrass theorem and to make sure that your algebra satisfies all of them. This may require some trial and error, but with persistence, you can definitely find an appropriate algebra for your problem. Good luck!