Banach Spaces as Data Types

TL;DR

This paper develops a framework for internal computability in Banach spaces using new operators, showing that externally computable functions are also internally computable, which simplifies understanding their complexity.

Contribution

It introduces the 'modified limit' and 'accumulation' operators on Banach spaces and proves the equivalence of external and internal computability for functions from metric to Banach spaces.

Findings

Externally computable functions are internally computable.

Internal computability simplifies complexity analysis.

Operators help define internal computability in Banach spaces.

Abstract

We introduce the operators "modified limit" and "accumulation" on a Banach space, and we use this to define what we mean by being internally computable over the space. We prove that any externally computable function from a computable metric space to a computable Banach space is internally computable. We motivate the need for internal concepts of computability by observing that the complexity of the set of finite sets of closed balls with a nonempty intersection is not uniformly hyperarithmetical, and thus that approximating an externally computable function is highly complex.