Applied Functional Analysis: Main Principles and Their Applications

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Springer Science & Business Media, Aug 30, 1995 - Mathematics - 406 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 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
References
371
List of Symbols
385
List of Theorems
391
List of the Most Important Definitions
393
Subject Index
399
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Funktionalanalysis
Dirk Werner
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