The Stone-Weierstrass Theorem

Suppose that $X$ is a topological space for which there is a collection of continuous real-valued functions on $X$ that separates points in $X$. Show that $X$ is Hausdorff.

Let $X$ be a compact, Hausdorff space and $\mathcal{A} \subseteq C(X)$ an algebra that contains the constant functions. Show that $\mathcal{A}$ is dense in $C(X)$ if and only if $\mathcal{A}$ separates points in $X$.

Let $\mathcal{A}$ be an algebra of continuous real-valued functions on a compact space $X$ that contains the constant functions. Let $f \in C(X)$ have the property that for some constant function $c$ and real number $\alpha$, the function $\alpha (f+c)$ belongs to $\overline{\mathcal{A}}$. Show that $f$ also belongs to $\overline{\mathcal{A}}$.

For $f, g \in C[a, b]$, show that $f = g$ if and only if $\int_{a}^{b} x^{n} f(x) d x = \int_{a}^{b} x^{n} g(x) d x$ for all $n$.

For $f \in C[a, b]$ and $\epsilon > 0$, show that there are real numbers $c_{0}, c_{1}, \ldots , c_{n}$ for which

For $f \in C[0, \pi ]$ and $\epsilon > 0$, show that there are real numbers $c_{0}, c_{1}, \ldots , c_{n}$ for which

Let $f$ be a continuous real-valued function on $\mathbb {R}$ that is periodic with period $2 \pi$. For $\epsilon > 0$, show that there are real numbers, $c_{0}, a_{1}, \ldots , a_{n}, b_{1}, \ldots , b_{n}$, such that

Let $X$ and $Y$ be compact, Hausdorff spaces and $f$ belong to $C(X \times Y)$. Show that for each $\epsilon > 0$, there are functions $f_{1}, \ldots , f_{n}$ in $C(X)$ and $g_{1}, \ldots , g_{n}$ in $C(Y)$ such that

Rather than use the Weierstrass Approximation Theorem in the proof of the Stone-Weierstrass Theorem, show that there are natural numbers $m$ and $n$ for which the polynomial $p(x) = 1-\left(1-x^{n}\right)^{m}$ satisfies (14).

Let $\mathcal{A}$ be a collection of continuous real-valued functions on a compact, Hausdorff space $X$ that separates the points of $X$. Show that every continuous real-valued function on $X$ can be uniformly approximated arbitrarily closely by a polynomial in a finite number of functions of $\mathcal{A}$.

Let $\mathcal{A}$ be an algebra of continuous real-valued functions on a compact, Hausdorff space $X$. Show that the closure of $\mathcal{A}, \overline{\mathcal{A}}$, also is an algebra.

Let $\mathcal{A}$ be an algebra of continuous real-valued functions on a compact, Hausdorff space $X$ that separates points. Show that either $\overline{\mathcal{A}} = C(X)$ or there is a point $x_{0} \in X$ for which $\overline{\mathcal{A}} = \left\{ f \in C(X) \mid f\left(x_{0}\right) = 0\right\}$.

Let $X$ be a compact, Hausdorff space and $\mathcal{A}$ an algebra of continuous functions on $X$ that separates points and contains the constant functions.