Introduction to Functional Analysis Theory and Applications

Introduction to Functional Analysis: Theory and Applications is a comprehensive and modern introduction to one of the most important foundations of contemporary mathematics. Originally published in Spanish and widely adopted throughout the Latin American academic community, this English edition presents a thoroughly revised and expanded translation of the acclaimed second edition.

Designed to serve both as a graduate-level textbook and a long-term reference work, the book develops the fundamental results of Functional Analysis while emphasizing their role in mathematics, physics, engineering, and computational science. Core topics include duality, linear operators, variational problems, compact operators, spectral theory, weak topologies, distribution theory, and Sobolev spaces, complemented by numerous applications to partial differential equations, numerical analysis, and finite element methods.

A distinguishing feature of the text is its unified and interconnected approach. Rather than treating subjects as isolated topics, the author carefully links concepts, proofs, and results across chapters, helping readers develop a deeper understanding of the discipline as a coherent whole. Banach and Hilbert space theory are intentionally woven together, enabling readers to appreciate both their common foundations and their essential differences. Likewise, important applied topics such as Babuška-Brezzi theory, distribution theory and Sobolev spaces are revisited throughout the book from multiple perspectives, reinforcing understanding and highlighting their broad relevance.

Drawing on decades of teaching and internationally recognized research, Gabriel N. Gatica combines rigorous mathematical exposition with practical applications, detailed proofs, and research-inspired exercises. The result is a distinctive and engaging text that supports advanced undergraduate and graduate study while providing researchers and practitioners with a valuable reference across a wide range of mathematical and engineering disciplines.

Ideal for courses in Functional Analysis, Applied Mathematics, Partial Differential Equations, Optimization, Numerical Analysis, Finite Element Methods, and Mathematical Methods in Physics and Engineering.

Gabriel N. Gatica is Full Professor in the Department of Mathematical Engineering and Senior Researcher at the Center for Research in Mathematical Engineering (CI²MA), University of Concepción, Chile. He is also an External Associate Researcher at the Center for Mathematical Modeling (CMM), University of Chile.

Professor Gatica is the author of more than 230 peer-reviewed research publications and six books. His research has made significant contributions to the development and analysis of innovative mixed finite element methods for a wide range of linear and nonlinear problems in continuum mechanics and electromagnetism. Over the past 25 years, he has published approximately 200 papers in this area, advancing both computational methods and their mathematical foundations. His work has also led to the establishment of new theoretical results whose impact extends well beyond his own research, influencing the broader fields of numerical analysis, partial differential equations, and applied mathematics.

Gabriel N. Gatica is recognized at both the national and international levels, as one of the main promoter of the development of Applied Mathematics, specially of Numerical Analysis of Partial Differential Equations, during the last 25 years in Chile, and particularly in Universidad de Concepción.