Computable copies of ℓp
Timothy H. McNicholl · Computability · 2016
Suppose p is a computable real so that [Formula: see text]. It is shown that the halting set can compute a surjective linear isometry between any two computable copies of [Formula: see text]. It is also shown that this result is optimal in that when [Formula: see text] there are two computable copies of [Formula: see text] with the property that any oracle that computes a linear isometry of one onto the other must also compute the halting set. Thus, [Formula: see text] is [Formula: see text]-categorical and is computably categorical if and only if [Formula: see text]. It is also demonstrated that there is a computably categorical Banach space that is not a Hilbert space. These results hold in both the real and complex case.