# Existence of bases for $L^2(X,\mu )$ with special properties.

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.

• Please ask one question per post. Feb 24 at 0:18
• 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.
– Ruy
Feb 24 at 0:53