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Let $X$ be a compact metric space and let $\mu $ be a Borel probability measure on $X$. The following questions regard the existence of orthonormal bases for the complex Hilbert space $L^2(X,\mu )$ with special properties.

  1. Does $L^2(X,\mu )$ admit a basis $\{f_i\}_i$ formed by continuous functions with $|f_i(x)|=1$, for every $i$ and $x$.

  2. Does $L^2(X,\mu )$ admit a basis $\{f_i\}_i$ with $|f_i(x)|=1$, for every $i$ and almost every $x$.

  3. Does $L^2(X,\mu )$ admit a basis $\{f_i\}_i$ formed by continuous functions with $\sup_{i, x}|f_i(x)|<\infty $.

  4. Does $L^2(X,\mu )$ admit a basis $\{f_i\}_i$ with $\sup_i\|f_i\|_\infty <\infty $.

The inexistence of atoms for $\mu$ might be relevant, so you are welcome to assume this in case it helps.


If you need motivation for this question, note that for every compact abelian group $G$, with normalized Haar measure, the answer to (1) is affirmative, with the group characters forming a basis.

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  • $\begingroup$ Please ask one question per post. $\endgroup$ Feb 24 at 0:18
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    $\begingroup$ Dear @Jose, please note that the questions are intimately related to each other and, in particular, they are listed in decreasing order of strength. A positive answer to question (1) implies positive answers to all other questions. It would certainly not make any sense to write four posts for what is essentially just one question. $\endgroup$
    – Ruy
    Feb 24 at 0:53

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