Which Lebesgue spaces are computably presentable?

Abstract: We consider the following question: ``If there is a computably presentable $L^p$ space, does it follow that $p$ is computable?” The answer is of course `no’ since the 1-dimensional $L^p$ space is just the field of scalars. So, we turn to non-trivial cases. Namely, assume there is a computably presentable $L^p$ space whose dimension is at least $2$. We prove $p$ is computable if the space is finite-dimensional or if $p \geq 2$. We then show that if $1 \leq p < 2$, and if $L^p[0,1]$ is computably presentable, then $p$ is right-c.e.. Finally, we show there is no uniform solution of this problem even when given upper and lower bounds on the exponent. The proof of this result leads to some basic results on the effective theory of stable random variables. Finally, we conjecture that the answer to this question is `no’ and that right-c.e.-ness of the exponent is the best result possible.

logic

Audience: researchers in the topic

Computability theory and applications

Series comments: Description: Computability theory, logic

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