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I am trying to prove the next proposition, but i need some help:

Let $ (X,\rho)$ be compact metric space. Prove that there exist a compact subset $K$ of $C(X)$ whose linear span is dense in $C(X)$

Thanks.

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    What is the linear span of $K$? I don't see any linear structure on $C(X)$.2012-12-28
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    @Alex: $C(X)$ is the space of continuous functions $X \to \mathbb{R}$...2012-12-28
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    @QiaochuYuan Ah, I forgot that convention. For some reason I was thinking of $C(X,X)$.2012-12-28
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    @Alex: Yes, $C(X)=C(X,R)$2012-12-29
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    Why are you trying to prove it? Are you sure it is true (i.e., is it an exercise from a book?)2012-12-29
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    You could try the set of all 1-Lipschitz functions bounded by 1. Use Arzelà-Ascoli and Stone-Weierstrass theorems.2012-12-29
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    @Thomas: is a qualifier problem.2012-12-29
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    @JulienB: What we get from the Stone-Weierstrass Theorem is that the $ \mathbb{R} $-subalgebra generated by the $ 1 $-Lipschitz functions is dense in $ C(X,\mathbb{R}) $. I do not see how we can deduce that the *$ \mathbb{R} $-linear span* of the $ 1 $-Lipschitz functions is dense in $ C(X,\mathbb{R}) $ from this result.2012-12-29

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