Elements of Functional Analysis

From Banach Spaces to Elliptic Problems
Erscheinungsjahr 2026
ISBN: 978-3-032-29209-4
Verlag: Springer

Buch, Englisch, 442 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 769 g

Reihe: Universitext

Functional analysis can be understood as a shift, within mathematical analysis, from the study of individual functions to the study of function spaces, their structures, and the mappings between them. Developed primarily during the 20th century, this theory continues to be essential for the study of partial differential equations.

This textbook begins with the general theory: Banach spaces, Hilbert spaces, and the spectral theory of operators. It then proceeds to a thorough account of Schwartz’s Theory of Distributions and some of its applications, such as the fundamental solutions of classical operators in physics, the Malgrange–Ehrenpreis Theorem, hypoelliptic operators, and the Schrödinger equation. An extensive chapter is dedicated to the study of Sobolev spaces, including functions of bounded variation. These spaces provide the appropriate framework for the study of elliptic boundary value problems, which form the focus of the final chapter.

Drawing on years of teaching experience, this textbook is ideal for introductory graduate courses, featuring historical notes and numerous exercises that reinforce learning and complement the core material.

Zielgruppe

Graduate

Autoren/Hrsg.

Mazón, José M.

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Funktionalanalysis
• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen

Inhaltsverzeichnis

Chapter 1. Brief Historical Notes.- Chapter 2. Banach Spaces.- Chapter 3. Hilbert Spaces.- Chapter 4. Spectral Theory of Operators.- Chapter 5. Theory of Distributions.- Chapter 6. Sobolev Spaces.- Chapter 7. Elliptic Boundary Value Problems.

Über Autor(innen)

José M. Mazón is an emertius professor of mathematical analysis at the University of Valencia. He is a specialist in nonlinear partial differential equations, a field in which he has published more than 160 research articles in prestigious journals. His research topics include singular and degenerate nonlinear PDEs, image processing equations, flux limited diffusion equations, mass transport theory and nonlocal equations. Some of his results have been published in six monographs: Parabolic Quasilinear Equations Minimizing Linear Growth Functional (Birkhäuser, winner of the 2023 Ferran Sunyer i Balaguer Prize), Nonlocal Diffusion Problems (American Mathematical Society, 2010), Nonlocal Perimeter, Curvature and Minimal Surfaces (Birkhäuser, 2019), Variational and Diffusion Problems in Random Walk Spaces (Birkhäuser, 2023), Functions of Least Gradient (Birkhäuser, 2024) and Weak Solutions to Gradient Flows in Metric Measure Spaces (Cambridge University Press, 2026).