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I need help with the following problem:

Let $(X,\rho)$ be a compact metric space. Prove that if $K$ is a compact subset of $C(X)=C(X,\mathbb{R})$ (i.e. continuous functions with real values) whose linear span is dense in $C(X)$, then the pseudometric $d(x,y)= \sup _{f \in K} \lvert f(x) - f(y)\rvert$ on $X$ is actually a metric. Moreover, show that $d$ and $\rho$ generate the same topology.

The existence of $K$ was proved in another thread here: Dense subset of $C(X)$.

Thanks.

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    Usually when you refer to another thread it makes sense to link to it. In the case of other threads on this site, you can just put the URL directly into the text and it will be replaced by the title of the post; for external URLs you can use the chain icon above the editing area to create links.2012-12-31
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    @Joriki: Sorry, I am just new here. Someone has done it already.2012-12-31
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    What topology do you assume on $C(X)$?2012-12-31
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    @ThomasE.: The OP is assuming that $ C(X,\mathbb{R}) $ is endowed with the sup-norm topology. I think he didn't specify it here because this problem is the sequel to another one whose thread link is provided above.2012-12-31

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