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Full Description
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.
Contents
The Projection Theorem.
Theorems on Extension and Separation.
Dual Spaces and Transposed Operators.
The Banach Theorem and the Banach-Steinhaus Theorem.
Construction of Hilbert Spaces.
L¯2 Spaces and Convolution Operators.
Sobolev Spaces of Functions of One Variable.
Some Approximation Procedures in Spaces of Functions.
Sobolev Spaces of Functions of Several Variables and the Fourier Transform.
Introduction to Set-Valued Analysis and Convex Analysis.
Elementary Spectral Theory.
Hilbert-Schmidt Operators and Tensor Products.
Boundary Value Problems.
Differential-Operational Equations and Semigroups of Operators.
Viability Kernels and Capture Basins.
First-Order Partial Differential Equations.
Selection of Results.
Exercises.
Bibliography.
Index.
