Analysis MOC

Stone-Weierstraß theorem

Let be a compact space and be a ∗-subalgebra of the continuous function ∗-algebra that is separating. Then is dense in , fun i.e. .

Proof

proof

Finite version

Let be a subalgebra of the function algebra on a finite set . Then iff separates points, linalg i.e. for any distinct there exists so that and

Proof

Assume is separating. For each , let so that ^[Invoking an Iverson bracket.]. Defining as above, it follows

and since span it follows . For the converse just set for all .

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