Applied Functional Analysis

D.H. Griffel

• Publisher: Dover Publications
• Year: 2012
• ISBN:
  9780486141329
• Topic: Mathematics
• Subtopic: Functional Analysis
• Index: Mathematics

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 edition. 25 Figures. 9 Appendices. Supplementary Problems. Indexes.

PART I

DISTRIBUTION THEORY AND GREEN’S FUNCTIONS

The central ideas of our subject, the theories of Banach and Hilbert spaces, are contained in Parts II and III. Part IV can be regarded as a coda, and Part I as a prelude. The theory of Parts II and III is independent of the distribution theory developed in Part I, which can be omitted if desired (though an acquaintance with Green’s functions would be helpful). In Part IV, however, the two theories will be unified.

The theory of distributions is essentially a new foundation for mathematical analysis; that is, a new structure, replacing functions of a real variable by new objects, defined quite differently, but having many of the same properties and usable for many of the same purposes as ordinary functions. This theory gives a useful technique for analysing linear problems in applied mathematics, but is less useful for nonlinear problems (which is why it is not used in Parts II and III). It leads naturally to the introduction of Green’s functions, which are essential to the application of functional analysis to differential equations. And it forms a useful prelude to Parts II and III because it introduces fairly abstract ideas in a fairly concrete setting.

Our treatment of distribution theory is intended to be detailed enough to give a good general understanding of the subject, but it does not aim at completeness. Many details and many worthwhile topics are omitted for the sake of digestibility. A full account would take a full-length book; see the references given in the last section of each chapter.

The plan of Part I is as follows. In Chapter 1 we set out the basic theory of distributions. Chapter 2 discusses ordinary differential equations from the distributional point of view, and introduces Green’s functions. Chapter 3 begins with a discussion of the Fourier transform from both the classical and distributional points of view, and then uses it to obtain Green’s functions for Laplace’s equation and for the wave equation.

The following sections form a short account of Green’s functions, which may be useful to readers who do not wish to learn distribution theory: 1.1; 2.3 as far as Example 2.9; 2.4; 2.5 omitting the proof of 2.15; 3.1; the second half of 3.4; 3.5 ; 3.6.

Chapter 1

Generalised Functions

In this chapter we lay the theoretical foundations for the treatment of differential equations in Chapters 2 and 3. We begin in section 1.1 by discussing the physical background of the delta function, which was the beginning of distribution theory. In section 1.2 we set out the basic theory of generalised functions or distributions (we do not distinguish between these terms), and in sections 1.3 and 1.4 we define the operations of algebra and calculus on generalised functions. The ideas and definitions of the theory are more elaborate than those of ordinary calculus; this is the price paid for developing a theory which is in many ways simpler as w...

Table of contents

• Title Page
• Copyright Page
• Table of Contents
• Preface
• PART I - DISTRIBUTION THEORY AND GREEN’S FUNCTIONS
• PART II - BANACH SPACES AND FIXED POINT THEOREMS
• PART III - OPERATORS IN HILBERT SPACE
• PART IV - FURTHER DEVELOPMENTS
• Appendices
• Notes on the Problems
• Supplementary Problems
• Index of Symbols
• References and Name Index
• Subject Index
• A CATALOG OF SELECTED DOVER BOOKS IN SCIENCE AND MATHEMATICS