Ljubiša D.R. Kočinac

Statistical Convergence in Function Spaces

We study statistical versions of several classical kinds of convergence of sequences of functions between metric spaces (Dini, Arzela, and Alexandroff) in different function spaces. Also, we discuss a statistical approach to recently introduced notions of strong uniform convergence and exhaustiveness.

Function spaces and a property of Reznichenko

Abstract In this paper we show that for a set X of real numbers the function space C p ( X ) has both a property introduced by Sakai in [Proc. Amer. Math. Soc. 104 (1988) 917–919] and a property introduced by Reznichenko (see [Topology Appl. 104 (2000) 181–190]) if and only if all finite powers of X have a property that was introduced by Gerlits and Nagy in [Topology Appl. 14 (1982) 151–161]. It follows that the minimal cardinality of a set of real numbers for which the function space does not have the properties of Sakai and Reznichenko is equal to the additivity of the ideal of first category sets of real numbers.

Double Sequences and Selections

We present some results about selection properties in the class of double sequences of real numbers.

On the class S 0 ¯ of real sequences

Abstract We prove that certain classes of sequences of positive real numbers satisfy some selection principles related to a special kind of convergence.

Summability of Sequences and Selection Properties

We prove that some classes of summable sequences of positive real numbers satisfy several selection principles related to a special kind of convergence.

Boundedness properties in function spaces

Abstract Some boundedness properties of function spaces (considered as topo- logical groups) are studied.

On a subclass of the class of rapidly varying sequences

We define and study the class KR ∞ , s which is a proper subclass of the class R ∞ , s of rapidly varying sequences of index of variability ∞. Then, we prove a theorem of the Galambos-Bojanic-Seneta type for this class, as well as a representation theorem in the Karamata sense. We also give some important asymptotic properties and characterizations of sequences in KR ∞ , s .

Some Topological Properties of Fuzzy Antinormed Linear Spaces

The definition of fuzzy antinorm is modified. Some topological properties of finite dimensional fuzzy antinormed linear space are studied. Fuzzy anticonvergence and statistical fuzzy anticonvergence are defined and their properties are studied. We also discuss some boundedness properties in fuzzy antinormed linear spaces.