Applied Functional Analysis: Applications to Mathematical Physics

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Springer Science & Business Media, Aug 13, 1999 - Mathematics - 481 pages
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A theory is the more impressive, the simpler are its premises, the more distinct are the things it connects, and the broader is its range of applicability. Albert Einstein There are two different ways of teaching mathematics, namely, (i) the systematic way, and (ii) the application-oriented way. More precisely, by (i), I mean a systematic presentation of the material governed by the desire for mathematical perfection and completeness of the results. In contrast to (i), approach (ii) starts out from the question "What are the most important applications?" and then tries to answer this question as quickly as possible. Here, one walks directly on the main road and does not wander into all the nice and interesting side roads. The present book is based on the second approach. It is addressed to undergraduate and beginning graduate students of mathematics, physics, and engineering who want to learn how functional analysis elegantly solves mathematical problems that are related to our real world and that have played an important role in the history of mathematics. The reader should sense that the theory is being developed, not simply for its own sake, but for the effective solution of concrete problems. viii Preface This introduction to functional analysis is divided into the following two parts: Part I: Applications to mathematical physics (the present AMS Vol. 108); Part II: Main principles and their applications (AMS Vol. 109).
 

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Contents

Banach Spaces and FixedPoint Theorems
1
11 Linear Spaces and Dimension
2
12 Normed Spaces and Convergence
7
13 Banach Spaces and the Cauchy Convergence Criterion
10
14 Open and Closed Sets
15
15 Operators
16
16 The Banach FixedPoint Theorem and the Iteration Method
18
17 Applications to Integral Equations
22
34 Applications to Polynomials
208
35 Unitary Operators
212
36 The Extension Principle
213
37 Applications to the Fourier Transformation
214
38 The Fourier Transform of Tempered Generalized Functions
219
Eigenvalue Problems for Linear Compact Symmetric Operators
229
41 Symmetric Operators
230
42 The HilbertSchmidt Theory
232

18 Applications to Ordinary Differential Equations
24
19 Continuity
26
110 Convexity
29
111 Compactness
33
112 FiniteDimensional Banach Spaces and Equivalent Norms
42
113 The Minkowski Functional and Homeomorphisms
45
114 The Brouwer FixedPoint Theorem
53
115 The Schauder FixedPoint Theorem
61
116 Applications to Integral Equations
62
117 Applications to Ordinary Differential Equations
63
118 The LeraySchauder Principle and a priori Estimates
64
119 Sub and Supersolutions and the Iteration Method in Ordered Banach Spaces
66
120 Linear Operators
70
121 The Dual Space
74
122 Infinite Series in Normed Spaces
76
124 Applications to Linear Differential Equations in Banach Spaces
80
125 Applications to the Spectrum
82
126 Density and Approximation
84
127 Summary of Important Notions
88
Hilbert Spaces Orthogonality and the Dirichlet Principle
101
21 Hilbert Spaces
105
22 Standard Examples
109
23 Bilinear Forms
120
24 The Main Theorem on Quadratic Variational Problems
121
25 The Functional Analytic Justification of the Dirichlet Principle
125
26 The Convergence of the Ritz Method for Quadratic Variational Problems
140
27 Applications to BoundaryValue Problems the Method of Finite Elements and Elasticity
145
28 Generalized Functions and Linear Functional
156
29 Orthogonal Projection
165
210 Linear Functionals and the Riesz Theorem
167
211 The Duality Map
169
213 The Linear Orthogonality Principle
172
214 Nonlinear Monotone Operators
173
215 Applications to the Nonlinear LaxMilgram Theorem and the Nonlinear Orthogonality Principle
174
Hilbert Spaces and Generalized Fourier Series
195
31 Orthonormal Series
199
32 Applications to Classic Fourier Series
203
33 The Schmidt Orthogonalization Method
207
43 The Fredholm Alternative
237
44 Applications to Integral Equations
240
45 Applications to BoundaryEigenvalue Problems
245
SelfAdjoint Operators the Friedrichs Extension and the Partial Differential Equations of Mathematical Physics
253
51 Extensions and Embeddings
260
52 SelfAdjoint Operators
263
53 The Energetic Space
273
54 The Energetic Extension
279
55 The Friedrichs Extension of Symmetric Operators
280
56 Applications to BoundaryEigenvalue Problems for the Laplace Equation
285
57 The Poincare Inequality and Rellichs Compactness Theorem
287
58 Functions of SelfAdjoint Operators
293
59 Semigroups OneParameter Groups and Their Physical Relevance
298
510 Applications to the Heat Equation
305
511 Applications to the Wave Equation
309
512 Applications to the Vibrating String and the Fourier Method
315
513 Applications to the Schrodinger Equation
323
514 Applications to Quantum Mechanics
327
515 Generalized Eigenfunctions
343
516 Trace Class Operators
347
517 Applications to Quantum Statistics
348
518 CAlgebras and the Algebraic Approach to Quantum Statistics
357
519 The Fock Space in Quantum Field Theory and the Pauli Principle
363
520 A Look at Scattering Theory
368
521 The Language of Physicists in Quantum Physics and the Justification of the Dirac Calculus
373
522 The Euclidean Strategy in Quantum Physics
379
523 Applications to Feynmans Path Integral
385
524 The Importance of the Propagator in Quantum Physics
394
525 A Look at Solitons and Inverse Scattering Theory
406
Epilogue
425
Appendix
429
References
443
Hints for Further Reading
457
List of Symbols
461
List of Theorems
467
List of the Most Important Definitions
469
Subject Index
473

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