Uniform domain representations of ℓp-spaces

✍ Scribed by Petter K. Køber

• Publisher: John Wiley and Sons
• Year: 2007
• Tongue: English
• Weight: 324 KB
• Volume: 53
• Category: Article
• ISSN: 0044-3050
• DOI: 10.1002/malq.200610039

✦ Synopsis

Abstract

The category of Scott‐domains gives a computability theory for possibly uncountable topological spaces, via representations. In particular, every separable Banach‐space is representable over a separable domain. A large class of topological spaces, including all Banach‐spaces, is representable by domains, and in domain theory, there is a well‐understood notion of parametrizations over a domain. We explore the link with parameter‐dependent collections of spaces in e. g. functional analysis through a case study of ℓ^p^ ‐spaces. We show that a well‐known domain representation of ℓ^p^ as a metric space can be made uniform in the sense of parametrizations of domains. The uniform representations admit lifting of continuous functions and are effective in p. Dependent type constructions apply, and through the study of the sum and product spaces, we clarify the notions of uniformity and uniform computability. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)