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Stone-Weierstrass Theorem

Stone-Weierstrass Theorem

Weierstrass Approximation theorem

Theorem 1 Weierstrass

There exists polynomial $p:[a, b] \rightarrow \mathbb{R}$ such that

\[\norm{ f - p }_{\infty} < \epsilon .\]

proof

$\Box$

Corollary 2

$C_{0}([a, b])$ is separable.

proof

Let

\[\begin{eqnarray} \mathcal{P}_{0} & := & \{ p(x) \mid p(x) = \sum_{k=0}^{n} a_{k}x^{k}, \ n \in \mathbb{Z}_{\ge 0}, \ a_{k} \in \mathbb{Q} \} \nonumber \\ \mathcal{P} & := & \{ p(x) \mid p(x) = \sum_{k=0}^{n} a_{k}x^{k}, \ n \in \mathbb{Z}_{\ge 0}, \ a_{k} \in \mathbb{R} \} \nonumber \end{eqnarray} .\]

Let $p = \sum_{k=0}^{n}c_{k}x^{k} \in \mathcal{P}$ and $\epsilon > 0$ be fixed. For all $k \in [0:n]$, there exists $d_{k} \in \mathbb{Q}$ such that

\[\begin{eqnarray} \abs{ c_{k} - d_{k} } & = & \frac{1}{b^{k}(n + 1)} \epsilon . \nonumber \end{eqnarray}\]

Let $q(x) := \sum_{k=0}^{n} d_{k}x^{k}$.

\[\begin{eqnarray} \abs{ p(x) - q(x) } & < & \sum_{k=0}^{n} \abs{c_{k} - d_{k}} \abs{x}^{k} \nonumber \\ & < & \sum_{k=0}^{n} \frac{\epsilon}{b^{k}(n + 1)} b^{k} \nonumber \\ & = & \sum_{k=0}^{n} \frac{\epsilon}{n + 1} \nonumber \\ & = & \epsilon \nonumber \end{eqnarray}\]

By Weierstrass approximation theorem, $\mathcal{P}{0}$ is also dense in $C([a, b])$. Besides, $\mathcal{P}{0}$ is countable.

$\Box$

Stone-Weirastrass theorem

Theorem 3 Stone-Weirastrass theorem

proof

$\Box$

Reference