Applied Functional Analysis

By: D. H. GRIFFEL

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Paperback | 14 June 2002

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390 Pages

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23.5 x 15.5 x 2.54

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A stimulating introductory text, this volume examines many important applications of functional analysis to mechanics, fluid mechanics, diffusive growth, and approximation. Detailed enough to impart a thorough understanding, the text is also sufficiently straightforward for those unfamiliar with abstract analysis. Its four-part treatment begins with distribution theory and discussions of Green's functions. Essentially independent of the preceding material, the second and third parts deal with Banach spaces, Hilbert space, spectral theory, and variational techniques. The final part outlines the ideas behind Frechet calculus, stability and bifurcation theory, and Sobolev spaces. 25 Figures. 9 Appendices. Supplementary Problems. Indexes.

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Non-Fiction Mathematics Calculus & Mathematical Analysis Functional Analysis & Transforms Applied Mathematics

Preface	p. 9
Distribution Theory and Green's Functions
Generalised Functions
The Delta Function	p. 12
Basic Distribution Theory	p. 15
Operations on Distributions	p. 19
Convergence of Distributions	p. 24
Further Developments	p. 26
Fourier Series and the Poisson Sum Formula	p. 30
Summary and References	p. 33
Problems	p. 33
Differential Equations and Green's Functions
The Integral of a Distribution	p. 37
Linear Differential Equations	p. 39
Fundamental Solutions of Differential Equations	p. 41
Green's Functions	p. 45
Applications of Green's Functions	p. 48
Summary and References	p. 51
Problems	p. 51
Fourier Transforms and Partial Differential Equations
The Classical Fourier Transform	p. 53
Distributions of Slow Growth	p. 56
Generalised Fourier Transforms	p. 60
Generalised Functions of Several Variables	p. 64
Green's Function for the Laplacian	p. 67
Green's Function for the Three-Dimensional Wave Equation	p. 74
Summary and References	p. 78
Problems	p. 78
Banach Spaces and Fixed Point Theorems
Normed Spaces
Vector Spaces	p. 84
Normed Spaces	p. 91
Convergence	p. 95
Open and Closed Sets	p. 98
Completeness	p. 104
Equivalent Norms	p. 110
Summary and References	p. 112
Problems	p. 112
The Contraction Mapping Theorem
Operators on Vector Spaces	p. 116
The Contraction Mapping Theorem	p. 120
Application to Differential and Integral Equations	p. 123
Nonlinear Diffusive Equilibrium	p. 128
Nonlinear Diffusive Equilibrium in Three Dimensions	p. 131
Summary and References	p. 134
Problems	p. 134
Compactness and Schauder's Theorem
Continuous Operators	p. 139
Brouwer's Theorem	p. 144
Compactness	p. 149
Relative Compactness	p. 152
Arzela's Theorem	p. 155
Schauder's Theorems	p. 158
Forced Nonlinear Oscillations	p. 160
Swirling Flow	p. 165
Summary and References	p. 170
Problems	p. 171
Operators in Hilbert Space
Hilbert Space
Inner Product Spaces	p. 176
Orthogonal Bases	p. 180
Orthogonal Expansions	p. 183
The Bessel, Parseval, and Riesz-Fischer Theorems	p. 188
Orthogonal Decomposition	p. 192
Functionals on Normed Spaces	p. 195
Functionals in Hilbert Space	p. 198
Weak Convergence	p. 199
Summary and References	p. 205
Problems	p. 206
The Theory of Operators
Bounded Operators on Normed Spaces	p. 210
The Algebra of Bounded Operators	p. 214
Self-Adjoint Operators	p. 221
Eigenvalue Problems for Self-Adjoint Operators	p. 226
Compact Operators	p. 230
Summary and References	p. 234
Problems	p. 235
The Spectral Theorem
The Spectral Theorem	p. 240
Sturm-Liouville Systems	p. 245
Partial Differential Equations	p. 252
The Fredholm Alternative	p. 257
Projection Operators	p. 260
Summary and References	p. 268
Problems	p. 269
Variational Methods
Positive Operators	p. 274
Approximation to the First Eigenvalue	p. 277
The Rayleigh-Ritz Method for Eigenvalues	p. 281
The Theory of the Rayleigh-Ritz Method	p. 284
Inhomogeneous Equations	p. 292
Complementary Bounds	p. 296
Summary and References	p. 301
Problems	p. 301
Further Developments
The Differential Calculus of Operators and its Applications
The Frechet Derivative	p. 308
Higher Derivatives	p. 312
Maxima and Minima	p. 316
Linear Stability Theory	p. 319
Nonlinear Stability	p. 323
Bifurcation Theory	p. 326
Bifurcation and Stability	p. 330
Summary and References	p. 334
Distributional Hilbert Spaces
The Space of Square-Integrable Distributions	p. 336
Sobolev Spaces	p. 340
Application to Partial Differential Equations	p. 343
Summary and References	p. 345
Appendices
Sets and Mappings	p. 347
Sequences, Series, and Uniform Convergence	p. 348
Sup and Inf	p. 352
Countability	p. 354
Equivalence Relations	p. 357
Completion	p. 357
Sturm-Liouville Systems	p. 360
Fourier's Theorem	p. 362
Proofs of 9.24 and 9.25	p. 364
Notes on the Problems	p. 367
Supplementary Problems	p. 375
Symbol Index	p. 379
References and Name Index	p. 382
Subject Index	p. 387
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Applied Functional Analysis

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Paperback

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