## Applied Functional Analysis: Applications to Mathematical PhysicsA 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 |

443 | |

457 | |

List of Symbols | 461 |

List of Theorems | 467 |

List of the Most Important Definitions | 469 |

473 | |

### Other editions - View all

Applied Functional Analysis: Applications to Mathematical Physics Eberhard Zeidler Limited preview - 2012 |

Applied Functional Analysis: Applications to Mathematical Physics Eberhard Zeidler No preview available - 1999 |

Applied Functional Analysis: Applications to Mathematical Physics Eberhard Zeidler No preview available - 1999 |

### Common terms and phrases

AMS Vol Applications Banach space boundary-value problem bounded called Cauchy sequence classical complete orthonormal system complex Hilbert space continuous functions convergent convex Corollary corresponds countable define Definition dense differential equations Dirichlet principle eigenfunctions eigenvalue energetic space energy equivalent exists Figure finite finite-dimensional fixed fixed-point theorem following hold true Formal Fourier series Fourier transformation Friedrichs extension functional analysis given Hence Hilbert space implies inner product integral equation Lebesgue Lemma linear continuous linear operator linear space linear subspace mathematics nonempty nonlinear normed space obtain open set orthogonal orthonormal system physicists physics Proof Proposition quantum mechanics real Hilbert space real numbers relatively compact Ritz method Schrodinger equation Schwarz inequality Section self-adjoint operator Sobolev spaces Standard Example Step subset Suppose symmetric operator theory u)un unique solution unitary variational problem yields