Functional analysis : an introduction to metric spaces, Hilbert spaces, and Banach algebras

Abstract

Originally, functional analysis was the study of functions. It is now considered to be a unifying subject that generalizes much of linear algebra and real/complex analysis, with emphasis on infinite dimensional spaces. This book introduces this vast topic from these elementary preliminaries and develops both the abstract theory and its applications in three parts: (I) Metric Spaces, (II) Banach and Hilbert Spaces, and (III) Banach Algebras. Especially with the digital revolution at the turn of the millennium, Hilbert spaces and least squares approximation have become necessary and fundamental topics for a mathematical education, not only just for mathematicians, but also for engineers, physicists, and statisticians interested in signal processing, data analysis, regression, quantum mechanics, etc. Banach spaces have gained popularity in applications and are complementing or even supplanting the classical least squares approach to many optimization problems.