Ultrafilters in Mathematics: An Overview

Ultrafilter appears as an utmost applicable concept in numerous branches of mathematics including logic, algebra, topology, nonstandard analysis, and functional analysis. In this article, regarding the properties and applications of ultrafilters, we intend to identify the structure of this concept specially in functional analysis. To this end, ultrafilters are investigated in terms of set theory, measure theory, Stone-Cech compactification, nonstandard analysis and functional analysis which enable us to describe this concept from various perspectives.