Let $\{e_n(x)\}_{n=1}^\infty$ be any orthonormal basis for $L^2[0,1]$, prove that $\sum_{n=1}^\infty|e_n(x)|^2$ diverges almost everywhere.
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Certainly not true for $\mathbb N$ with counting measure. – Robert Israel Jan 16 '18 at 22:07
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@RobertIsrael Thanks! I delete the last sentence. – Liding Yao Jan 16 '18 at 23:48
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1Possible duplicate of $\sum |f_n (x)|https://math.stackexchange.com/questions/249464 – Kavi Rama Murthy Feb 07 '18 at 10:23
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@KaviRamaMurthy You're right – Liding Yao Feb 07 '18 at 16:05