## Normann, Dag - Banach Spaces as Data Types

lmcs:1018 - Logical Methods in Computer Science, May 17, 2011, Volume 7, Issue 2
Banach Spaces as Data Types

Authors: Normann, Dag

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.

Source : oai:arXiv.org:1104.5307
DOI : 10.2168/LMCS-7(2:11)2011
Volume: Volume 7, Issue 2
Published on: May 17, 2011
Submitted on: November 3, 2010
Keywords: Mathematics - Logic,F.1.1, F.4.1, F.3.2