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Can you show that if $f \in C[0,1]$, and $\int_{0}^{1} f x^p dx =0$ for all primes $p$, that $f \equiv 0$?

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up vote 10 down vote accepted

The key point is Müntz-Szasz theorem, which states that for a sequence $(\lambda_n)_{n\geqslant 1}$ of positive numbers, the vector space generated by the constant functions and $\{x\mapsto x^{\lambda_n},n\geqslant 1\}$ is dense in $C[0,1]$ endowed with the uniform norm if and only if $\sum_{n\geqslant 1}1/\lambda_n$ is divergent. Then we conclude.

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Do you know of a good online source for the proof of that theorem? I hadn't heard of that one before. – Johnny Apple May 1 '14 at 22:01
@JohnnyApple, It seems that there is a proof in Rudin's RCA. Search this site for Szasz. – lhf May 1 '14 at 22:08

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