Computable copies of $\ell^p$

Abstract

\begin{abstract} Suppose $p$ is a computable real so that $p \geq 1$. It is shown that the halting set can compute a surjective linear isometry between any two computable copies of $\ell^p$. It is also shown that this result is optimal in that when $p \neq 2$ there are two computable copies of $\ell^p$ with the property that any oracle that computes a linear isometry of one onto the other must also compute the halting set. Thus, $\ell^p$ is $\Delta_2^0$-categorical and is computably categorical if and only if $p = 2$. It is also shown that there is a computably categorical Banach space that is not a Hilbert space and that $\ell^p$ is linearly isometric to a computable Banach space if and only if $p$ is computable. These results hold in both the real and complex case.