## Applied Functional Analysis: Main Principles and Their ApplicationsA 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 Our introduction to applied functional analysis is divided into two parts: Part I: Applications to Mathematical Physics (AMS Vol. 108); Part II: Main Principles and Their Applications (AMS Vol. 109). A detailed discussion of the contents can be found in the preface to AMS Vol. 108. |

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### Contents

The HahnBanach Theorem and Optimization Problems | 1 |

11 The HahnBanach Theorem | 2 |

12 Applications to the Separation of Convex Sets | 6 |

13 The Dual Space Cab | 10 |

14 Applications to the Moment Problem | 13 |

15 Minimum Norm Problems and Duality Theory | 15 |

16 Applications to Cebysev Approximation | 19 |

17 Applications to the Optimal Control of Rockets | 20 |

311 The Exactness of the Duality Functor | 205 |

312 Applications to the Closed Range Theorem and to Predholm Alternatives | 210 |

The Implicit Function Theorem | 225 |

41 mLinear Bounded Operators | 227 |

42 The Differential of Operators and the Frechet Derivative | 228 |

43 Applications to Analytic Operators | 233 |

44 Integration | 238 |

45 Applications to the Taylor Theorem | 243 |

Variational Principles and Weak Convergence | 39 |

21 The 7ith Variation | 43 |

22 Necessary and Sufficient Conditions for Local Extrema and the Classical Calculus of Variations | 45 |

InfiniteDimensional Banach Spaces | 48 |

24 Weak Convergence | 49 |

25 The Generalized Weierstrass Existence Theorem | 53 |

26 Applications to the Calculus of Variations | 56 |

27 Applications to Nonlinear Eigenvalue Problems | 59 |

28 Reflexive Banach Spaces | 61 |

29 Applications to Convex Minimum Problems and Variational Inequalities | 66 |

210 Applications to Obstacle Problems in Elasticity | 71 |

211 Saddle Points | 72 |

212 Applications to Duality Theory | 73 |

213 The von Neumann Minimax Theorem on the Existence of Saddle Points | 75 |

214 Applications to Game Theory | 81 |

215 The Ekeland Principle about QuasiMinimal Points | 83 |

Principle via the PalaisSmale Condition | 86 |

217 Applications to the Mountain Pass Theorem | 87 |

218 The Galerkin Method and Nonlinear Monotone Operators | 93 |

219 Symmetries and Conservation Laws The Noether Theorem | 98 |

220 The Basic Ideas of Gauge Field Theory | 102 |

221 Representations of Lie Algebras | 107 |

222 Applications to Elementary Particles | 112 |

Principles of Linear Functional Analysis | 167 |

31 The Baire Theorem | 169 |

Nondifferentiable Continuous Functions | 171 |

33 The Uniform Boundedness Theorem | 172 |

34 Applications to Cubature Formulas | 175 |

35 The Open Mapping Theorem | 178 |

36 Product Spaces | 180 |

37 The Closed Graph Theorem | 181 |

38 Applications to Factor Spaces | 183 |

39 Applications to Direct Sums and Projections | 188 |

310 Dual Operators | 199 |

46 Iterated Derivatives | 244 |

47 The Chain Rule | 247 |

48 The Implicit Function Theorem | 250 |

49 Applications to Differential Equations | 254 |

410 Diffeomorphisms and the Local Inverse Mapping Theorem | 258 |

411 Equivalent Maps and the Linearization Principle | 260 |

412 The Local Normal Form for Nonlinear Double Splitting Maps | 264 |

413 The Surjective Implicit Function Theorem | 268 |

414 Applications to the Lagrange Multiplier Rule | 270 |

Fredholm Operators | 281 |

51 Duality for Linear Compact Operators | 284 |

52 The RieszSchauder Theory on Hilbert Spaces | 286 |

53 Applications to Integral Equations | 291 |

54 Linear Fredholm Operators | 292 |

55 The RieszSchauder Theory on Banach Spaces | 295 |

56 Applications to the Spectrum of Linear Compact Operators | 296 |

57 The Parametrix | 298 |

58 Applications to the Perturbation of Fredholm Operators | 300 |

59 Applications to the Product Index Theorem | 301 |

510 Fredholm Alternatives via Dual Pairs | 303 |

511 Applications to Integral Equations and BoundaryValue Problems | 305 |

512 Bifurcation Theory | 309 |

513 Applications to Nonlinear Integral Equations | 313 |

514 Applications to Nonlinear BoundaryValue Problems | 315 |

515 Nonlinear Fredholm Operators | 317 |

516 Interpolation Inequalities | 322 |

517 Applications to the NavierStokes Equations | 329 |

371 | |

List of Symbols | 385 |

List of Theorems | 391 |

List of the Most Important Definitions | 393 |

399 | |

### Other editions - View all

Applied Functional Analysis: Main Principles and Their Applications Eberhard Zeidler Limited preview - 2012 |

Applied Functional Analysis: Main Principles and Their Applications Eberhard Zeidler No preview available - 1995 |

Applied Functional Analysis: Main Principles and Their Applications Eberhard Zeidler No preview available - 1995 |

### Common terms and phrases

AMS Vol Applications bifurcation bijective boundary bounded called Cauchy classical closed linear subspace codim condition continuous functional convex sets Corollary corresponds Define definition denotes dense derivative Differential Equations dim N(A direct sum dual space duality embedding equivalent exists a point F-derivative F(un Figure Fredholm operators functional F Furthermore Gateaux derivative given Hahn-Banach theorem Hence Hilbert space implicit function theorem implies inequality inverse mapping theorem Lemma Lie algebra linear continuous functional linear continuous operator linear operator linear space linear subspace lower semicontinuous mathematical minimum problem Moreover nonempty nonlinear normed space obtain open neighborhood open set operator equation outer force densities particle principle proof of Proposition proof of Theorem prove real numbers reflexive Banach space respect sequence Sobolev spaces Standard Example Step subset sufficiently small Suppose surjective theorem Theorem topological space variational problem vector weakly