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Applied Functional Analysis

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Applied Functional Analysis Cover

 

Synopses & Reviews

Publisher Comments:

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. 1985 ed. 25 Figures. 9 Appendices. Supplementary Problems. Indexes.

Book News Annotation:

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 (booknews.com)

Synopsis:

This introductory text examines applications of functional analysis to mechanics, fluid mechanics, diffusive growth, and approximation. Covers distribution theory, Banach spaces, Hilbert space, spectral theory, Frechet calculus, Sobolev spaces, more. 1985 edition.

Synopsis:

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.

Table of Contents

  Preface

Part I. Distribution Theory and Green's Functions

  Chapter 1. Generalised Functions

    1.1 The Delta function

    1.2 Basic distribution theory

    1.3 Operations on distributions

    1.4 Convergence of distributions

    1.5 Further developments

    1.6 Fourier Series and the Poisson Sum formula

    1.7 Summary and References

    Problems

  Chapter 2. Differential Equations and Green's Functions

    2.1 The Integral of a distribution

    2.2 Linear differential equations

    2.3 Fundamental solutions of differential equations

    2.4 Green's functions

    2.5 Applications of Green's functions

    2.6 Summary and References

    Problems

  Chapter 3. Fourier Transforms and Partial differential Equations

    3.1 The classical Fourier transform

    3.2 Distributions of slow growth

    3.3 Generalised Fourier transforms

    3.4 Generalised functions of several variables

    3.5 Green's function for the Laplacian

    3.6 Green's function for the Three-dimensional wave equation

    3.7 Summary and References

    Problems

Part II. Banach spaces and fixed point theorems

  Chapter 4. Normed spaces

    4.1 Vector spaces

    4.2 Normed spaces

    4.3 Convergence

    4.4 Open and closed sets

    4.5 Completeness

    4.6 Equivalent norms

    4.7 Summary and References

    Problems

  Chapter 5. The contraction mapping theorem

    5.1 Operators on Vector spaces

    5.2 The contraction mapping theorem

    5.3 Application to differential and integral equations

    5.4 Nonlinear diffusive equilibrium

    5.5 Nonlinear diffusive equilibrium in three dimensions

    5.6 Summary and References

    Problems

  Chapter 6. Compactness and Schauder's theorem

    6.1 Continuous operators

    6.2 Brouwer's theorem

    6.3 Compactness

    6.4 Relative compactness

    6.5 Arzelà's theorem

    6.6 Schauder's theorems

    6.7 Forced nonlinear oscillations

    6.8 Swirling flow

    6.9 Summary and References

    Problems

Part III. Operators in Hilbert Space

  Chapter 7. Hilbert space

    7.1 Inner product spaces

    7.2 Orthogonal bases

    7.3 Orthogonal expansions

    7.4 The Bessel, Parseval, and Riesz-Fischer theorems

    7.5 Orthogonal decomposition

    7.6 Functionals on normed spaces

    7.7 Functionals in Hilbert space

    7.8 Weak convergence

    7.9 Summary and References

    Problems

  Chapter 8. The Theory of operators

    8.1 Bounded operators on normed spaces

    8.2 The algebra of bounded operators

    8.3 Self-adjoint operators

    8.4 Eigenvalue problems for self-adjoint operators

    8.5 Compact operators

    8.6 Summary and References

    Problems

  Chapter 9. The Spectral theorem

    9.1 The spectral theorem

    9.2 Sturm-Liouville systems

    9.3 Partial differential equations

    9.4 The Fredholm alternative

    9.5 Projection operators

    9.6 Summary and References

    Problems

  Chapter 10. Variational methods

    10.1 Positive operators

    10.2 Approximation to the first eigenvalue

    10.3 The Rayleigh-Ritz method for eigenvalues

    10.4 The theory of the Rayleigh-Ritz method

    10.5 Inhomogeneous Equations

    10.6 Complementary bounds

    10.7 Summary and References

    Problems

Part IV. Further developments

  Chapter 11. The differential calculus of operators and its applications

    11.1 The Fréchet derivative

    11.2 Higher derivatives

    11.3 Maxima and Minima

    11.4 Linear stability theory

    11.5. Nonlinear stability

    11.6 Bifurcation theory

    11.7 Bifurcation and stability

    11.8 Summary and References

  Chapter 12. Distributional Hilbert spaces

    12.1 The space of square-integrable distributions

    12.2 Sobolev spaces

    12.3 Application to partial differential equations

    12.4 Summary and References

Appendix A. Sets and mappings

Appendix B. Sequences, series, and uniform convergence

Appendix C. Sup and inf

Appendix D. Countability

Appendix E. Equivalence relations

Appendix F. Completion

Appendix G. Sturm-Liouville systems

Appendix H. Fourier's theorem

Appendix I. Proofs of 9.24 and 9.25

  Notes on the Problems; Supplementary Problems; Symbol index; References and name index; Subject index

Product Details

ISBN:
9780486422589
Author:
Griffel, D. H.
Publisher:
Dover Publications
Author:
Mathematics
Location:
Mineola, N.Y.
Subject:
Calculus
Subject:
Applied
Subject:
Functional Analysis
Subject:
General Mathematics
Subject:
Mathematics : Functional Analysis
Edition Number:
Dover ed.
Edition Description:
Trade Paper
Series:
Dover Books on Mathematics
Series Volume:
no 95
Publication Date:
20020631
Binding:
TRADE PAPER
Language:
English
Illustrations:
25 Figures
Pages:
400
Dimensions:
9.25 x 6.13 in 1.18 lb

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Related Subjects

History and Social Science » World History » General
Science and Mathematics » Mathematics » Applied
Science and Mathematics » Mathematics » Calculus » General
Science and Mathematics » Mathematics » Functional Analysis
Science and Mathematics » Mathematics » General

Applied Functional Analysis New Trade Paper
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Product details 400 pages Dover Publications - English 9780486422589 Reviews:
"Synopsis" by ,
This introductory text examines applications of functional analysis to mechanics, fluid mechanics, diffusive growth, and approximation. Covers distribution theory, Banach spaces, Hilbert space, spectral theory, Frechet calculus, Sobolev spaces, more. 1985 edition.
"Synopsis" by ,
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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