A novel, practical introduction to functional analysis
In the twenty years since the first edition of Applied Functional Analysis was published, there has been an explosion in the number of books on functional analysis. Yet none of these offers the unique perspective of this new edition. Jean-Pierre Aubin updates his popular reference on functional analysis with new insights and recent discoveries-adding three new chapters on set-valued analysis and convex analysis, viability kernels and capture basins, and first-order partial differential equations. He presents, for the first time at an introductory level, the extension of differential calculus in the framework of both the theory of distributions and set-valued analysis, and discusses their application for studying boundary-value problems for elliptic and parabolic partial differential equations and for systems of first-order partial differential equations.
To keep the presentation concise and accessible, Jean-Pierre Aubin introduces functional analysis through the simple Hilbertian structure. He seamlessly blends pure mathematics with applied areas that illustrate the theory, incorporating a broad range of examples from numerical analysis, systems theory, calculus of variations, control and optimization theory, convex and nonsmooth analysis, and more. Finally, a summary of the essential theorems as well as exercises reinforcing key concepts are provided. Applied Functional Analysis, Second Edition is an excellent and timely resource for both pure and applied mathematicians.Content:
Chapter 1 The Projection Theorem (pages 4โ26):
Chapter 2 Theorems on Extension and Separation (pages 27โ48):
Chapter 3 Dual Spaces and Transposed Operators (pages 49โ69):
Chapter 4 The Banach Theorem and the Banach?Steinhaus Theorem (pages 70โ93):
Chapter 5 Construction of Hilbert Spaces (pages 94โ119):
Chapter 6 L2 Spaces and Convolution Operators (pages 120โ144):
Chapter 7 Sobolev Spaces of Functions of One Variable (pages 145โ166):
Chapter 8 Some Approximation Procedures in Spaces of Functions (pages 167โ186):
Chapter 9 Sobolev Spaces of Functions of Several Variables and the Fourier Transform (pages 187โ210):
Chapter 10 Introduction to Set?Valued Analysis and Convex Analysis (pages 211โ257):
Chapter 11 Elementary Spectral Theory (pages 259โ282):
Chapter 12 Hilbert?Schmidt Operators and Tensor Products (pages 283โ308):
Chapter 13 Boundary Value Problems (pages 309โ359):
Chapter 14 Differential?Operational Equations and Semigroups of Operators (pages 360โ384):
Chapter 15 Viability Kernels and Capture Basins (pages 385โ410):
Chapter 16 First?Order Partial Differential Equations (pages 411โ447):