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I was looking for a simple application of the Stone-Weierstrass theorem.

First I thought that if $X$ is any compact measure space then Stone-Weierstrass implies that $C_c(X)$ is dense in $L^p$.

But I have to assume that $X$ is compact otherwise I don't have $1$ in $C_c(X)$. That of course makes it a boring example since then $C_c(X) = C(X)$. Can someone show me a slightly more interesting but still simple example? Thank you.

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    Well, one standard application of the theorem is that the trigonometric polynomials are dense in $C([0,2\pi])$, say. Note that this is not immediate from Fourier Analyis (only if you consider Cesaro summation).2012-07-13
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    I don't see how Stone-Weierstrass implies the density of $C_c(X)$ in $L^p$, since this theorem only gives density with respect to the supremum norm.2012-07-13
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    @DavideGiraudo Doh! *facepalm* Thank you!2012-07-13
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    See http://math.stackexchange.com/questions/107837/equality-of-measures2012-07-13
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    @MattN. The Stone-Weierstrass theorem implies that any commutative $C^*$-algebra is equivalent to $C_0(X)$ for some locally compact $X$.2012-07-13

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