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arXiv · 1104.5307

Banach Spaces as Data Types

Dag Normann
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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.

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Dag Normann. 2011-04-28. Banach Spaces as Data Types. https://doi.org/10.2168/lmcs-7(2:11)2011

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