# Continuous functions on $J\times J\rightarrow \mathbb{R}$ can be uniformly approached by continuous functions: $f_1(x) g_1(y)+…+f_n(x) g_n(y)$

Let $$J\subseteq \mathbb{R}$$ be a compact interval and let $$\mathbb{A}$$ be a collection of continuous functions on $$J \rightarrow \mathbb{R}$$ which satisfy the properties of the Stone-Weierstrass theorem.

Show that any continuous function on $$J\times J$$ (in $$\mathbb{R^2}$$) to $$\mathbb{R}$$ can be uniformly approximated by functions of the form: $$f_1(x) g_1(y)+f_2(x) g_2(y)+...f_n(x) g_n(y)$$ where $$f_i,g_i$$ belong to $$\mathbb{A}$$.

Note: this is exercise 26H of Bartle´s Elements of Real Analysis.

• What do you mean by "satisfy the properties of the Stone-Weierstrass Theorem"? And you can probably use the SW theorem with the collection of functions of the form $f_1g_1+\dots+f_ng_n$ (where $f_i,g_i\in\mathbb{A}$); although it's not obvious that this is closed under multiplication without assuming something about $\mathbb{A}$. – Reveillark Oct 14 '19 at 3:33
• do you mean $f_1(x) g_1(y) + \cdots + f_n(x) g_n(y)$? – user125932 Oct 14 '19 at 4:03
• Properties of SW are: constant function exists, it is closed under multiplication, it separates points and if $f,g \in \mathbb{A}$ then $\alpha f+\beta g \in \mathbb{A}$ – PLanderos33 Oct 14 '19 at 4:19

The Stone–Weierstraß-Theorem says that if $$X$$ is a compact Hausdorff space and $$\mathcal A$$ is a set of continuous functions $$X \to \mathbb R$$ which has the Stone-Weierstraß-property (SW-property) for $$X$$ as defined in your comment, then each continuous $$f : X \to \mathbb R$$ can be uniformly approximated by functions in $$\mathcal A$$.

So let $$X_i$$ be compact Hausdorff spaces and $$\mathcal A_i$$ be sets with the SW-property for $$X_i$$. Let $$\mathcal A_1 \times \mathcal A_2 = \{ f g \mid f \in \mathcal A_1, g \in \mathcal A_2 \}$$ and be $$\mathcal A$$ be the set of finite sums of elements of $$\mathcal A_1 \times \mathcal A_2$$.

Then $$\mathcal A$$ has the SW-property for $$X_1 \times X_2$$:

1. Let $$c \in \mathcal A_1, d \in \mathcal A_2$$ be constant and non-zero. Then $$c d \in \mathcal A$$ is constant and non-zero.

2. Let $$fg \in\mathcal A_1 \times \mathcal A_2$$ and $$\alpha \in \mathbb R$$. Then $$\alpha (fg) = (\alpha f)g \in \mathcal A_1 \times \mathcal A_2$$. Therefore, if $$u \in \mathcal A$$, then also $$\alpha u \in \mathcal A$$.

3. If $$u, v \in \mathcal A$$, then $$u + v$$ is trivially contained in $$\mathcal A$$.

4. If $$\sum_{i=1}^n f_ig_i, \sum_{j=1}^m f'_jg'_j\in \mathcal A$$, then $$(\sum_{i=1}^n f_ig_i) \cdot (\sum_{j=1}^m f'_jg'_j) = \sum_{i=1}^n (\sum_{j=1}^m (f_ig_i)(f'_jg'_j)) = \sum_{i=1}^n (\sum_{j=1}^m (f_if'_j)(g_ig'_j)) \in \mathcal A .$$

5. Let $$(x_1,x_2), (y_1,y_2) \in X_1 \times X_2$$ be two didtinct points. Then $$x_1 \ne y_1$$ or $$x_2 \ne y_2$$. In the first case there exists $$f \in \mathcal A_1$$ such that $$f(x_1) \ne f(y_1)$$. Choose a constant non-zero $$d \in \mathcal A_2$$. Then $$\phi = fc \in \mathcal A$$ and $$\phi(x_1,x_2) = cf(x_1) \ne cf(y_1) = \phi(y_1,y_2)$$. The other case is similar.