Let $(X,d)$ be a compact metric space and let $C(X)$ be the set of continuous (bounded) real-valued functions on $X$ equipped with the usual supremum norm: $$ \|f\|_{\infty}\triangleq \sup_{x\in X}|f(x)|. $$ Apparently, under these conditions $C(X)$ admits a Schauder basis. In general, how can we construct explicitly such Schauder bases?
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$\begingroup$ Semadeni published a SLN on this topic. Presumably you can find what you want there (I don't have access at the moment). $\endgroup$– bathalf15320Commented Jul 28, 2021 at 6:49
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$\begingroup$ What is an SLN? $\endgroup$– John_AlgorithmCommented Jul 28, 2021 at 7:02
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3$\begingroup$ Presumably referring to these Springer Lecture Notes. $\endgroup$– James TenerCommented Jul 28, 2021 at 7:11
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2$\begingroup$ Yes, thanks--I thought this was common knowledge amongst maths researchers. @John_Algorithm By the way, if you are happy to confine your attention to uncountable spaces, you can use Milyutin's theorem (just google) to reduce to the $C([0,1])$ case, where an explicit basis (actually THE Schauder basis) was already known to the polish school (see Banach's book). $\endgroup$– bathalf15320Commented Jul 28, 2021 at 7:53
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3$\begingroup$ math.stackexchange.com/questions/667251/… $\endgroup$– Bill JohnsonCommented Jul 28, 2021 at 15:12
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