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On regulated functions

100%

Cichoń K. , Cichoń M. , Satco B.

2018

tom Nr 60

37--57

In this paper we investigate the space of regulated functions on a compact interval [0,1]. When equipped with the topology of uniform convergence this space is isometrically isomorphic to some space of continuous functions. We study some of its properties, including the characterization of the dual space, weak and strong compactness properties of sets. Finally, we investigate some compact and weakly compact operators on the space of regulated functions. The paper is complemented by an existence result for the Hammerstein-Stieltjes integral equation with regulated solutions.

On regulated functions

63%

Banaś J. , Kot M.

Journal of Mathematics and Applications

2017

tom Vol. 40

21--36

The main purpose of this review article is to present the concept of a regulated function and to indicate the connection of the class of regulated functions with other significant classes of functions. In particular, we give a characterization of regulated functions in terms of step functions and we show that the linear space of regulated functions forms a Banach space under the classical supremum norm.

On Banach spaces of regulated functions

63%

Drewnowski L.

tom Vol 57, No. 2

153--169

For a relatively compact subset S of the real line R, let R(S) denote the Banach space (under the sup norm) of all regulated scalar functions defined on S. The purpose of this paper is to study those closed subspaces of R(S) that consist of functions that are left-continuous, right-continuous, continuous, and have a (two-sided) limit at each point of some specified disjoint subsets of S. In particular, some of these spaces are represented as C(K) spaces for suitable, explicitly constructed, compact spaces K.