Banach Spaces as Data Types

Details Banach Spaces as Data Types Banach Spaces as Data Types

2011-04-28

arxiv.org/abs/1104.5307v2

LMCS 7 (2:11) 2011

Mathematics

Logic

20 pages

Scientific paper

10.2168/LMCS-7(2:11)2011

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.

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