A separation form of unified convergence theorem

Abstract

In this paper, by introducing isometrically Pc 0 property a separation form of convergence theorem is presented and the results generalize and unify several interesting conclusions in recent years. §1 Introduction In [1], Qu and Wu pointed out an equivalent form of the famous Antosik-Mikusinski theorem. In [2], Aizpuru improved Antosik-Mikusinski theorem by using a separation property of natural families and established its equivalent form. Recently, Wu proved an excellent unified convergence theorem which presents five equivalent forms of Antosik-Mikusinski theorems [3] . In this paper, moreover, we show a new convergence theorem which generalizes Wu and Aizpuru's interesting conclusions. is an Abelian topological group, then the topology τ can be generated by a family of quasi-norms [4] . So, without loss of generality, here we only consider the Abelian quasi-normed groups.We say that F is a natural family if φ 0 (N) ⊆F⊆ P (N), where φ 0 (N) and P (N) denote the families of all finite subsets and all subsets of N, respectively.A natural family F is said to have the property P c0 if there exists a map g : N → N such that for every pair of sequences (j r ) r and (m r ) r in N with j 1 < m 1 < j 2 < m 2 < · · · there exist B ∈ F and an infinite subset M of N such that (i) (m r−1 , m r ) B = {j r } for every r ∈ M, r > 1; (ii) card([m r−1 , m r ] B) ≤ g(r) for every r ∈ N \ M . Let (G, p) be an Abelian quasi-normed group. is isometric P c0 -Cauchy matrix if there exists a map g : N → N such that if (j r ) r and (m r ) r