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.

  • $\begingroup$ 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 $\mathbb{A}$. $\endgroup$ – Reveillark Oct 14 '19 at 3:33
  • 1
    $\begingroup$ do you mean $f_1(x) g_1(y) + \cdots + f_n(x) g_n(y)$? $\endgroup$ – user125932 Oct 14 '19 at 4:03
  • $\begingroup$ Properties of SW are: constant function exists, it is closed under multiplication, it separates points and if $f,g \in \mathbb{A}$ then $\alpha f+\beta g \in \mathbb{A}$ $\endgroup$ – 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.


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