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From Sets and Types to Topology and AnalysisTowards practicable foundations for constructive mathematics$
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Laura Crosilla and Peter Schuster

Print publication date: 2005

Print ISBN-13: 9780198566519

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198566519.001.0001

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COMPUTABILITY ON NON-SEPARABLE BANACH SPACES AND LANDAU'S THEOREM

COMPUTABILITY ON NON-SEPARABLE BANACH SPACES AND LANDAU'S THEOREM

Chapter:
(p.316) 21 COMPUTABILITY ON NON-SEPARABLE BANACH SPACES AND LANDAU'S THEOREM
Source:
From Sets and Types to Topology and Analysis
Author(s):

Vasco Brattka

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198566519.003.0021

In computable analysis, a computable normed space turns out to be necessarily separable. In classical functional analysis, non-separable spaces occur naturally, for example as dual spaces of common separable spaces. The question arises whether the notion of a computable normed space can be generalized to the non-separable case in a meaningful way. This chapter proves that there are at least two concepts which capture this notion, one of which is stronger than the other. The first focuses on computability of the norm, while the second one preserves computability of the linear operations. One can measure the quality of these concepts by regarding the number of classical results which can be proved computationally by applying the corresponding concept. Using the first concept, the chapter proves a computable version of Landau's Theorem for sequence spaces, which suggests that priority should be given to the computability of the norm.

Keywords:   computable analysis, functional analysis, normed space, non-separable space, Landau's Theorem

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