An Application: The Stone-Weierstrass Theorem

Suppose that $\mathcal{A}$ is a subalgebra of $C(X)$ that separates the points of $X$ and $1 \in \mathcal{A}$. Show that if $x_1, \dots , x_n$ are distinct points in $X$ and $\alpha_1, \dots , \alpha_n \in \mathbb {F}$, there is an $f$ in $\mathcal{A}$ such that $f(x_j) = \alpha_j$ for $1 \leq j \leq n$.

Give the details of the proof of Corollary 8.3: if $X$ is locally compact and $\mathcal{A}$ is a closed subalgebra of $C_0(X)$ such that

If $X$ is compact, show that for each $x$ in $X$, $\delta_x$ is an extreme point of $P(X)$ and $\alpha \delta_x$, $\left|\alpha \right| = 1$, is an extreme point of $\ball M(X)$.

Let $X$ be compact and let $\mathcal{A}$ be a closed subalgebra of $C(X)$ such that $1 \in \mathcal{A}$ and $\mathcal{A}$ is closed under conjugation. Define an equivalence relation $\sim$ on $X$ by declaring $x \sim y$ if and only if $f(x) = f(y)$ for all $f$ in $\mathcal{A}$. Let $X / \sim$ be the corresponding quotient space and let $\pi : X \to X / \sim$ be the natural map. Give $X / \sim$ the quotient topology.

Let $X$ be completely regular and topologize $C(X)$ as in Example IV.1.5. If $\mathcal{A}$ is a closed subalgebra of $C(X)$ such that $1 \in \mathcal{A}$, $\mathcal{A}$ separates the points of $X$, and $\bar{f} \in \mathcal{A}$ whenever $f \in \mathcal{A}$, then $\mathcal{A} = C(X)$.

Let $X, Y$ be compact spaces and show that if $f \in C(X \times Y)$ and $\varepsilon > 0$, then there are functions $g_1, \dots , g_n$ in $C(X)$ and $h_1, \dots , h_n$ in $C(Y)$ such that $\left|f(x,y) - \sum_{k=1}^n g_k(x) h_k(y)\right| < \varepsilon$ for all $(x,y)$ in $X \times Y$.

Let $\mathcal{A}$ be the uniformly closed subalgebra of $C_b(\mathbb {R})$ generated by $\sin x$ and $\cos x$. Show that $\mathcal{A} = \left\{ f \in C_b(\mathbb {R}) : f(t) = f(t + 2\pi ) \text{ for all } t \text{ in } \mathbb {R} \right\}$.

If $K$ is a compact subset of $\mathbb {C}$, $f \in C(K)$, and $\varepsilon > 0$, show that there is a polynomial $p(z, \bar{z})$ in $z$ and $\bar{z}$ such that $\left|f(z) - p(z,\bar{z})\right| < \varepsilon$ for all $z$ in $K$.