Applied Functional Analysis
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This introductory text examines many important applications of functional analysis to mechanics, fluid mechanics, diffusive growth, and approximation. Discusses distribution theory, Green's functions, Banach spaces, Hilbert space, spectral theory, and variational techniques. Also outlines the ideas behind Frechet calculus, stability and bifurcation theory, and Sobolev spaces. 1985 edition. Includes 25 figures and 9 appendices. Supplementary problems. Indexes. Booknews This textbook studies distribution theory and Green's functions, Banach spaces and fixed point theorems, and operators in Hilbert spaces. Griffel (mathematics, University of Bristol, UK) supplies applications in fluid mechanics, approximation, and dynamical systems. This is an unabridged reprint of the 1985 revised edition published by Ellis Horwood. Annotation c. Book News, Inc., Portland, OR
Preface9Part IDistribution Theory and Green's FunctionsChapter 1Generalised Functions1.1The Delta Function121.2Basic Distribution Theory151.3Operations on Distributions191.4Convergence of Distributions241.5Further Developments261.6Fourier Series and the Poisson Sum Formula301.7Summary and References33Problems33Chapter 2Differential Equations and Green's Functions2.1The Integral of a Distribution372.2Linear Differential Equations392.3Fundamental Solutions of Differential Equations412.4Green's Functions452.5Applications of Green's Functions482.6Summary and References51Problems51Chapter 3Fourier Transforms and Partial Differential Equations3.1The Classical Fourier Transform533.2Distributions of Slow Growth563.3Generalised Fourier Transforms603.4Generalised Functions of Several Variables643.5Green's Function for the Laplacian673.6Green's Function for the Three-Dimensional Wave Equation743.7Summary and References78Problems78Part IIBanach Spaces and Fixed Point TheoremsChapter 4Normed Spaces4.1Vector Spaces844.2Normed Spaces914.3Convergence954.4Open and Closed Sets984.5Completeness1044.6Equivalent Norms1104.7Summary and References112Problems112Chapter 5The Contraction Mapping Theorem5.1Operators on Vector Spaces1165.2The Contraction Mapping Theorem1205.3Application to Differential and Integral Equations1235.4Nonlinear Diffusive Equilibrium1285.5Nonlinear Diffusive Equilibrium in Three Dimensions1315.6Summary and References134Problems134Chapter 6Compactness and Schauder's Theorem6.1Continuous Operators1396.2Brouwer's Theorem1446.3Compactness1496.4Relative Compactness1526.5Arzela's Theorem1556.6Schauder's Theorems1586.7Forced Nonlinear Oscillations1606.8Swirling Flow1656.9Summary and References170Problems171Part IIIOperators in Hilbert SpaceChapter 7Hilbert Space7.1Inner Product Spaces1767.2Orthogonal Bases1807.3Orthogonal Expansions1837.4The Bessel, Parseval, and Riesz-Fischer Theorems1887.5Orthogonal Decomposition1927.6Functionals on Normed Spaces1957.7Functionals in Hilbert Space1987.8Weak Convergence1997.9Summary and References205Problems206Chapter 8The Theory of Operators8.1Bounded Operators on Normed Spaces2108.2The Algebra of Bounded Operators2148.3Self-Adjoint Operators2218.4Eigenvalue Problems for Self-Adjoint Operators2268.5Compact Operators2308.6Summary and References234Problems235Chapter 9The Spectral Theorem9.1The Spectral Theorem2409.2Sturm-Liouville Systems2459.3Partial Differential Equations2529.4The Fredholm Alternative2579.5Projection Operators2609.6Summary and References268Problems269Chapter 10Variational Methods10.1Positive Operators27410.2Approximation to the First Eigenvalue27710.3The Rayleigh-Ritz Method for Eigenvalues28110.4The Theory of the Rayleigh-Ritz Method28410.5Inhomogeneous Equations29210.6Complementary Bounds29610.7Summary and References301Problems301Part IVFurther DevelopmentsChapter 11The Differential Calculus of Operators and its Applications11.1The Frechet Derivative30811.2Higher Derivatives31211.3Maxima and Minima31611.4Linear Stability Theory31911.5Nonlinear Stability32311.6Bifurcation Theory32611.7Bifurcation and Stability33011.8Summary and References334Chapter 12Distributional Hilbert Spaces12.1The Space of Square-Integrable Distributions33612.2Sobolev Spaces34012.3Application to Partial Differential Equations34312.4Summary and References345AppendicesASets and Mappings347BSequences, Series, and Uniform Convergence348CSup and Inf352DCountability354EEquivalence Relations357FCompletion357GSturm-Liouville Systems360HFourier's Theorem362IProofs of 9.24 and 9.25364Notes on the Problems367Supplementary Problems375Symbol Index379References and Name Index382Subject Index387
