A Unified Convergence Theorem

Abstract

We prove a unified convergence theorem, which presents in four equivalent forms of the famous Antosik-Mikusinski Theorems. In particular, we show that Swartz' three uniform convergence principles are all equivalent to the Antosik-Mikusinski Theorems.

A separation form of unified convergence theorem

“…Now, using Lemma 2.1 and the proof methods of [3], we can present the following theorem, for simplicity, we omit its proof. Theorem 2.2.…”

“…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.…”

On the Mikusiński-Antosik diagonal theorem and the equivalence of two types of convergence in Köthe spaces

“…These and other abstract forms of the sliding-hump method, including Rosenthal's lemma [25], the Antosik-Swartz basic matrix theorem [9,33], Antosik's lemma [6, 37], Weber's lemma [36], were studied by many authors; see e.g. [1][2][3]15,23,35,[37][38][39][40] and the references in [9] and [33]. It should be also noticed that the Mikusiński-Antosik diagonal theorem is related to the famous Ramsey theorem (see [24] and [23]) whose various versions and generalizations have been investigated and applied in varied areas for many years by numerous authors (see e.g.…”

“…The diagonal type theorems stand for a very useful tool in proving numerous theorems in measure theory and functional analysis formulated in a more general way than their previous versions proved by means of the Baire category method; see the articles [4,6,20,[35][36][37][38]40], the monographs [9,33] and references there.…”