ON $\sigma$-LIMIT AND $s\sigma$-LIMIT IN BANACH SPACES

Limit points of arithmetic means of sequences in Banach spaces

whenever a belongs to the closed convex hull of the set of weak limit points of {an}∞n=1. In case X has the Banach-Saks property and {an}∞n=1 is bounded the converse assertion holds too. A characterization of reflexive spaces in terms of limit points and cores of bounded sequences is also given. The motivation for the problems investigated goes back to Lévy laplacian from potential theory in Hi...

On Ω-limit Sets of Ordinary Differential Equations in Banach Spaces

We classify ω-limit sets of autonomous ordinary differential equations x′ = f(x), x(0) = x0, where f is Lipschitz, in infinite dimensional Banach spaces as being of three types I-III. Let S ⊂ X be a Polish subset of a Banach space X. S is of type I if there exists a Lipschitz function f and a solution x such that S = Ω(x) and x ∩ S = ∅. S is of type II if it has non-empty interior and there exi...

Limit spaces with approximations

Limit spaces and transfinite types

We give a characterisation of an extension of the Kleene-Kreisel continuous functionals to objects of transfinite types using limit spaces of transfinite types.

On the Relationship between Filter Spaces and Weak Limit Spaces

Countably based filter spaces have been suggested in the 1970’s as a model for recursion theory on higher types. Weak limit spaces with a countable base are known to be the class of spaces which can be handled by the Type-2 Model of Effectivity (TTE). We prove that the category of countably based proper filter spaces is equivalent to the category of countably based weak limit spaces. This resul...