Applied Functional Analysis

by Aubin, Jean-Pierre

• ISBN13: 9780471179764
• ISBN10: 0471179760
• Edition: 2nd
• Format: Hardcover
• Copyright: 2000-02-04
• Publisher: Wiley-Interscience

Summary

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.

Author Biography

JEAN-PIERRE AUBIN, PhD, is a professor at the UniversitT Paris-Dauphine in Paris, France. A highly respected member of the applied mathematics community, Jean-Pierre Aubin is the author of sixteen mathematics books on numerical analysis, neural networks, game theory, mathematical economics, nonlinear and set-valued analysis, mutational analysis, and viability theory.

Table of Contents

Preface	xiii
Introduction: A Guide to the Reader	1	(3)
The Projection Theorem	4	(23)
Definition of a Hilbert Space	4	(6)
Review of Continuous Linear and Bilinear Operations	10	(3)
extension of Continuous Linear and Bilinear Operators by Density	13	(2)
The Best Approximation Theorem	15	(3)
Orthogonal Projectors	18	(4)
Closed Subspaces, Quotient Spaces, and Finite Products of Hilbert Spaces	22	(1)
Orthogonal Bases for a Separable Hilbert Space	23	(4)
Theorems on Extension and Separation	27	(22)
Extension of Continuous Linear and Bilinear Operators	28	(1)
A Density Criterion	29	(1)
Separation Theorems	30	(2)
A Separation Theorem in Finite Dimensional Spaces	32	(1)
Support Functions	32	(2)
The Duality Theorem in Convex Optimization	34	(5)
Von Neumann's Minimax Theorem	39	(6)
Characterization of Pareto Optima	45	(4)
Dual Spaces and Transposed Operators	49	(21)
The Dual of a Hilbert Space	50	(4)
Realization of the Dual of a Hilbert Space	54	(2)
Transposition of Operators	56	(1)
Transposition of Injective Operators	57	(3)
Duals of Finite Products, Quotient Spaces, and Closed or Dense Subspaces	60	(4)
The Theorem of Lax-Milgram	64	(1)
Variational Inequalities	65	(2)
Noncooperative Equilibria in n-Person Quadratic Games	67	(3)
The Banach Theorem and the Banach-Steinhaus Theorem	70	(24)
Properties of Bounded Sets of Operators	71	(5)
The Mean Ergodic Theorem	76	(3)
The Banach Theorem	79	(3)
The Closed Range Theorem	82	(2)
Characterization of Left Invertible Operators	84	(2)
Characterization of Right Invertible Operators	86	(4)
Quadratic Programming with Linear Constraints	90	(4)
Construction of Hilbert Spaces	94	(26)
The Initial Scalar Product	96	(2)
The Final Scalar Product	98	(1)
Normal Subspaces of a Pivot Space	99	(5)
Minimal and Maximal Domains of a Closed Family of Operators	104	(3)
Unbounded Operators and Their Adjoints	107	(3)
Completion of a Pre-Hilbert Space Contained in a Hilbert Space	110	(1)
Hausdorff Completion	111	(1)
The Hilbert Sum of Hilbert Spaces	112	(3)
Reproducing Kernels of a Hilbert Space of Functions	115	(5)
L2 Spaces and Convolution Operators	120	(25)
The Space L2 (ω) of Square Integrable Functions	121	(3)
The Spaces L2 (ω, a) with Weights	124	(1)
The Space Hs	125	(3)
The Convolution Product for Functions of 0 (Rn) and of L1 (Rn)	128	(3)
Convolution Operators	131	(2)
Approximation by Convolution	133	(2)
Example. Convolution Power for Characteristic Functions	135	(4)
Example. Convolution Product for Polynomials: Appell Polynomials	139	(6)
Sobolev Spaces of Functions of One Variable	145	(22)
The Space Hm0 (ω) and Its Dual H--m (ω)	146	(2)
Definition of Distributions	148	(1)
Differentiation of Distributions	149	(4)
Relations Between Hm0 (ω) and Hm0 (R)	153	(1)
The Sobolev Space Hm (ω)	154	(4)
Relations Between Hm (ω) and Hm (R)	158	(3)
Characterization of the Dual of Hm (ω)	161	(2)
Trace Theorems	163	(1)
Convolution of Distributions	164	(3)
Some Approximation Procedures in Spaces of Functions	167	(20)
Approximation by Orthogonal Polynomials	168	(2)
Legendre, Laguerre, and Hermite Polynomials	170	(3)
Fourier Series	173	(2)
Approximation by Step Functions	175	(2)
Approximation by Piecewise Polynomial Functions	177	(6)
Approximation in Sobolev Spaces	183	(4)
Sobolev Spaces of Functions of Several Variables and the Fourier Transform	187	(24)
The Sobolev Spaces Hm0 (ω), Hm (ω), and H--m (ω)	188	(2)
The Fourier Transform of Infinitely Differentiable and Rapidly Decreasing Functions	190	(6)
The Fourier Transform of Sobolev Spaces	196	(3)
The Trace Theorem for the Spaces Hm (Rn+)	199	(7)
The Trace Theorem for the Spaces Hm (ω)	206	(3)
The Compactness Theorem	209	(2)
Introduction to Set-Valued Analysis and Convex Analysis	211	(48)
Graphical Derivations	213	(4)
Jumps Maps of Vector Distributions	217	(5)
Epiderivatives	222	(8)
Dual Concepts	230	(4)
Conjugate Functions	234	(16)
Economic Optima	250	(9)
Elementary Spectral Theory	259	(24)
Compact Operators	260	(2)
The Theory of Riesz-Fredholm	262	(4)
Characterization of Compact Operators from One Hilbert Space to Another	266	(2)
The Fredholm Alternative	268	(3)
Applications: Constructions of Intermediate Spaces	271	(3)
Application: Best Approximation Processes	274	(5)
Perturbation of an Isomorphism by a Compact Operator	279	(4)
Hilbert-Schmidt Operators and Tensor Products	283	(26)
The Hilbert Space of Hilbert-Schmidt Operators	284	(8)
The Fundamental Isomorphism Theorem	292	(1)
Hilbert Tensor Products	293	(5)
The Tensor Product of Continuous Linear Operators	298	(4)
The Hilbert Tensor Product by l2	302	(1)
The Hilbert Tensor Product by L2	303	(3)
The Tensor Product by the Sobolev Space Hm	306	(3)
Boundary Value Problems	309	(51)
The Formal Adjoint of an Operator and Green's Formula	312	(9)
Green's Formula for Bilinear Forms	321	(6)
Abstract Variational Boundary Value Problems	327	(8)
Examples of Boundary Value Problems	335	(6)
Approximation of Solutions to Neumann Problems	341	(5)
Restriction and Extension of the Formal Adjoint	346	(5)
Unilateral Boundary Value Problems	351	(3)
Introduction to Calculus of Variations	354	(6)
Differential-Operational Equations and Semigroups of Operators	360	(25)
Semigroups of Operators	362	(5)
Characterization of Infinitesimal Generators of Semigroups	367	(5)
Differential-Operational Equations	372	(3)
Boundary Value Problems for Parabolic Equations	375	(2)
Systems Theory: Internal and External Representations	377	(8)
Viability Kernels and Capture Basins	385	(26)
The Nagumo Theorem	386	(13)
Viability Kernels and Capture Basins	399	(12)
First-Order Partial Differential Equations	411	(37)
Some Hamilton-Jacobi Equations	414	(14)
Systems of First-Order Partial Differential Equations	428	(6)
Lotka-McKendrick Systems	434	(11)
Distributed Boundary Data	445	(3)
Selection of Results	448	(22)
General Properties	448	(2)
Properties of Continuous Linear Operators	450	(1)
Separation Theorems and Polarity	451	(1)
Construction of Hilbert Spaces	452	(2)
Compact Operators	454	(2)
Semigroup of Operators	456	(1)
The Green's Formula	456	(1)
Set-Valued Analysis and Optimization	457	(2)
Convex Analysis	459	(4)
Minimax Inequalities	463	(1)
Sobolev Spaces, Convolution, and Fourier Transform	463	(2)
Viability Kernels and Capture Basins	465	(2)
First-Order Partial Differential Equations	467	(3)
Exercises	470	(18)
Bibliography	488	(5)
Index	493