Introduction to Functional Analysis

This textbook offers an accessible introduction to Functional Analysis, providing a solid foundation for students new to the field. It is designed to support learners with no prior background in the subject and serves as an effective guide for introductory courses, suitable for students in mathematics and other STEM disciplines.

The book provides a comprehensive introduction to the essential topics of Functional Analysis across the first seven chapters, with a particular emphasis on normed vector spaces, Banach spaces, and continuous linear operators. It examines the parallels and distinctions between Functional Analysis and Linear Algebra, highlighting the crucial role of continuity in infinite-dimensional spaces and its implications for complex mathematical problems.

Later chapters broaden the scope, including advanced topics such as topological vector spaces, techniques in Nonlinear Analysis, and key theorems in theory of Banach spaces. Exercises throughout the book reinforce understanding and allow readers to test their grasp of the material.

Designed for students in mathematics and other STEM disciplines, as well as researchers seeking a thorough introduction to Functional Analysis, this book takes a clear and accessible approach. Prerequisites include a strong foundation in analysis in the real line, linear algebra, and basic topology, with helpful references provided for additional consultation.

Contents.- Preface.- Normed Vector Spaces.- Continuous Linear Operators.- Hahn-Banach Theorems.- Duality and Reflexive Spaces.- Hilbert Spaces.- Weak Topologies.- Spectral Theories of Compact Self-Adjoint Operators.- Topological Vector Spaces.- Introduction to Nonlinear Analysis.- Elements of Banach Space Theory.- A Zorn’s Lemma.- B Concepts of General Topology.- C Measure and Integration.- D Answers/hints for selected exercises.- Bibliography.- Index.

Geraldo Botelho is a Full Professor at the Federal University of Uberlândia, Brazil. He earned his PhD in 1995 from the State University of Campinas, Brazil, and has undertaken post-doctoral studies at the Universidad de Valencia, Spain (2006-2007), as well as a visiting period at the Universitat Potsdam, Germany in 1998.

Daniel Pellegrino is a Full Professor at the Federal University of Paraíba, Brazil. He obtained his PhD from the State University of Campinas, Brazil, in 2002. In 2020, Dr. Pellegrino was elected as a permanent Fellow of the Brazilian Academy of Sciences. Currently, he serves as the Editor-in-Chief of the Bulletin of the Brazilian Mathematical Society, published by Springer.

Eduardo Teixeira is a Full Professor at the University of Central Florida, USA. Prior to this, he held the same position at the Federal University of Ceará, Brazil. He earned his PhD in 2005 from the University of Texas at Austin, USA. In 2013, Dr. Teixeira was awarded the Mathematical Congress of the Americas Prize in 2013, elected permanent Fellow of the Brazilian Academy of Sciences in 2015, and in 2017 received the ICTP-IMU Ramanujan Prize for his significant contributions to the area of partial differential equations.