A Unified Convergence Theorem

TL;DR

This paper presents a unified convergence theorem that consolidates four equivalent forms of the Antosik-Mikusinski Theorems and demonstrates the equivalence of Swartz's uniform convergence principles with these theorems.

Contribution

The paper introduces a unified convergence theorem that links multiple forms of the Antosik-Mikusinski Theorems and related uniform convergence principles.

Findings

Four equivalent forms of the Antosik-Mikusinski Theorems are established.

Swartz's three uniform convergence principles are shown to be equivalent to these theorems.

The unified theorem simplifies understanding of convergence principles in analysis.

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.