Uniform domain representations of ℓ^p‐spaces

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 p -spaces. We show that a well-known domain representation of 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.

Reducibility of domain representations and Cantor–Weihrauch domain representations

“…Other lifting results where density of D is not required can be found in Normann (2001), Blanck (2001) and Køber (2007).…”

“…A more systematic study of representations from ordered structures, as well as the notion of domain representation, originate with Stoltenberg-Hansen and Tucker (1985;1988;1991;1993;1995;2007). Continuous domain representations are used in Edalat (1995a;1995b;, Edalat and Heckmann (1998) and Edalat and Lieutier (1999; to cover application areas from integration and differential equations to solid geometry.…”

Computability on Topological Spaces via Domain Representations

“…One way to deal with this problem is to use partial continuous functions [20]. There are important situations where liftings can be achieved also for non-dense representations [14,45,56]. Like the definition of domain representability this notion of reducibility works with many types of ordered structures.…”