Sen R. - A First Course in Functional Analysis_ Theory and Applications-Anthem Press (2013).pdf

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I feel I owe an explanation as to why I should write a new book, when a large number of books on functional analysis at the elementary level are available. Behind every abstract thought there is a concrete structure. I have tried to unveil the motivation behind every important development of the subject matter. I have endeavoured to make the presentation lucid and simple so that the learner can read without outside help. The first chapter, entitled ‘Preliminaries’, contains discussions on topics of which knowledge will be necessary for reading the later chapters. The first concepts introduced are those of a set, the cardinal number, the different operations on a set and a partially ordered set respectively. Important notions like Zorn’s lemma, Zermelo’s axiom of choice are stated next. The concepts of a function and mappings of different types are introduced and exhibited with examples. Next comes the notion of a linear space and examples of different types of linear spaces. The definition of subspace and the notion of linear dependence or independence of members of a subspace are introduced. Ideas of partition of a space as a direct sum of subspaces and quotient space are explained. ‘Metric space’ as an abstraction of real line is introduced. A broad overview of a metric space including the notions of convergence of a sequence, completeness, compactness and criterion for compactness in a metric space is provided in the first chapter. Examples of a non-metrizable space and an incomplete metric space are also given. The contraction mapping principle and its application in solving different types of equations are demonstrated. The concepts of an open set, a closed set and an neighbourhood in a metric space are also explained in this chapter. The necessity for the introduction of ‘topology’ is explained first. Next, the axioms of a topological space are stated. It is pointed out that the conclusions of the Heine-Borel theorem in a real space are taken as the axioms of an abstract topological space. Next the ideas of openness and closedness of a set, the neighbourhood of a point in a set, the continuity of a mapping, compactness, criterion for compactness and separability of a space naturally follow. Chapter 2 is entitled ‘Normed Linear Space’. If a linear space admits a metric structure it is called a metric linear space. A normed linear space is a type of metric linear space, and for every element x of the space there exists a positive number called norm x or x fulfilling certain axioms. A normed linear space can always be reduced to a metric space by the choice of a suitable metric. Ideas of convergence in norm and completeness of a normed linear space are introduced with examples of several normed linear spaces, Banach spaces (complete normed linear spaces) and incomplete normed linear spaces. 4 vii Continuity of a norm and equivalence of norms in a finite dimensional normed linear space are established. The definition of a subspace and its various properties as induced by the normed linear space of which this is a subspace are discussed. The notion of a quotient space and its role in generating new Banach spaces are explained. Riesz’s lemma is also discussed. Chapter 3 dwells on Hilbert space. The concepts of inner product space, complete inner product or Hilbert space are introduced. Parallelogram law, orthogonality of vectors, the Cauchy-Bunyakovsky-Schwartz inequality, and continuity of scalar (inner) product in a Hilbert space are discussed. The notions of a subspace, orthogonal complement and direct sum in the setting of a Hilbert space are introduced. The orthogonal projection theorem takes a special place. Orthogonality, various orthonormal polynomials and Fourier series are discussed elaborately. Isomorphism between separable Hilbert spaces is also addressed. Linear operators and their elementary properties, space of linear operators, linear operators in normed linear spaces and the norm of an operator are discussed in Chapter 4. Linear functionals, space of bounded linear operators and the uniform boundedness principle and its applications, uniform and pointwise convergence of operators and inverse operators and the related theories are presented in this chapter. Various types of linear operators are illustrated. In the next chapter, the theory of linear functionals is discussed. In this chapter I introduce the notions of a linear functional, a bounded linear functional and the limiting process, and assert continuity in the case of boundedness of the linear functional and vice-versa. In the case of linear functionals apart from different examples of linear functionals, representation of functionals in different Banach and Hilbert spaces are studied. The famous Hahn-Banach theorem on the extension on a functional from a subspace to the entire space with preservation of norm is explained and the consequences of the theorem are presented in a separate chapter. The notions of adjoint operators and conjugate space are also discussed. Chapter 6 is entitled ‘Space of Bounded Linear Functionals’. The chapter dwells on the duality between a normed linear space and the space of all bounded linear functionals on it. Initially the notions of dual of a normed linear space and the transpose of a bounded linear operator on it are introduced. The zero spaces and range spaces of a bounded linear operator and of its duals are related. The duals of Lp ([a, b]) and C([a, b]) are described. Weak convergence in a normed linear space and its dual is also discussed. A reflexive normed linear space is one for which the canonical embedding in the second dual is surjective (one-toone). An elementary proof of Eberlein’s theorem is presented. Chapter 7 is entitled ‘Closed Graph Theorem and its Consequences’. At the outset the definitions of a closed operator and the graph of an operator are given. The closed graph theorem, which establishes the conditions under which a closed linear operator is bounded, is provided. After introducing the concept of an viii open mapping, the open mapping theorem and the bounded inverse theorem are proved. Application of the open mapping theorem is also provided. The next chapter bears the title ‘Compact Operators on Normed Linear Spaces’. Compact linear operators are very important in applications. They play a crucial role in the theory of integral equations and in various problems of mathematical physics. Starting from the definition of compact operators, the criterion for compactness of a linear operator with a finite dimensional domain or range in a normed linear space and other results regarding compact linear operators are established. The spectral properties of a compact linear operator are studied. The notion of the Fredholm alternative is discussed and the relevant theorems are provided. Methods of finding an approximate solution of certain equations involving compact operators in a normed linear space are explored. Chapter 9 bears the title ‘Elements of Spectral Theory on Self-adjoint Operators in Hilbert Spaces’. Starting from the definition of adjoint operators, self-adjoint operators and their various properties are elaborated upon the context of a Hilbert space. Quadratic forms and quadratic Hermitian forms are introduced in a Hilbert space and their bounds are discovered. I define a unitary operator in a Hilbert space and the situation when two operators are said to be unitarily equivalent, is explained. The notion of a projection operator in a Hilbert space is introduced and its various properties are investigated. Positive operators and the square root of operators in a Hilbert space are introduced and their properties are studied. The spectrum of a self-adjoint operator in a Hilbert space is studied and the point spectrum and continuous spectrum are explained. The notion of invariant subspaces in a Hilbert space is also brought within the purview of the discussion. Chapter 10 is entitled ‘Measure and Integration in Spaces’. In this chapter I discuss the theory of Lebesgue integration and p-integrable functions on . Spaces of these functions provide very useful examples of many theorems in functional analysis. It is pointed out that the concept of the Lebesgue measure is a generalization of the idea of subintervals of given length in to a class of subsets in . The ideas of the Lebesgue outer measure of a set E ⊂ , Lebesgue measurable set E and the Lebesgue measure of E are introduced. The notions of measurable functions and integrable functions in the sense of Lebesgue are explained. Fundamental theorems of Riemann integration and Lebesgue integration, Fubini and Toneli’s theorem, are stated and explained. Lp spaces (the space of functions p-integrable on a measure subset E of ) are introduced, that (E) is complete and related properties discussed. Fourier series and then Fourier integral for functions are investigated. In the next chapter, entitled ‘Unbounded Linear Operators’, I first give some examples of differential operators that are not bounded. But these are closed operators, or at least have closed linear extensions. It is indicated in this chapter that many of the important theorems that hold for continuous linear operators on a Banach space also hold for closed linear operators. I define the different states of an operator depending on whether 4 4 4 4 ix 4 the range of the operator is the whole of a Banach space or the closure of the range is the whole space or the closure of the range is not equal to the space. Next the characterization of states of operators is presented. Strictly singular operators are then defined and accompanied by examples. Operators that appear in connection with the study of quantum mechanics also come within the purview of the discussion. The relationship between strictly singular and compact operators is explored. Next comes the study of perturbation theory. The reader is given an operator ‘A’, the certain properties of which need be found out. If ‘A’ is a complicated operator, we sometimes express ‘A = T +B’ where ‘T ’ is a relatively simple operator and ‘B’ is related to ‘T ’ in such a manner that knowledge about the properties of ‘T ’ is sufficient to gain information about the corresponding properties of ‘A’. In that case, for knowing the specific properties of ‘A’, we can replace ‘A’ with ‘T ’, or in other words we can perturb ‘A’ by ‘T ’. Here we study perturbation by a bounded linear operator and perturbation by strictly singular operator. Chapter 12 bears the title ‘The Hahn-Banach Theorem and the Optimization Problems’. I first explain an optimization problem. I define a hyperplane and describe what is meant by separating a set into two parts by a hyperplane. Next the separation theorems for a convex set are proved with the help of the Hahn-Banach theorem. A minimum Norm problem is posed and the Hahn-Banach theorem is applied to the proving of various duality theorems. Said theorem is applied to prove Chebyshev approximation theorems. The optimal control problem is posed and the Pontryagin’s problem is mentioned. Theorems on optimal control of rockets are proved using the Hahn-Banach theorem. Chapter 13 is entitled ‘Variational Problems’ and begins by introducing a variational problem. The aim is to investigate under which conditions a given functional in a normed linear space admits of an optimum. Many differential equations are often difficult to solve. In such cases a functional is built out of the given equation and minimized. One needs to show that such a minimum solves the given equation. To study those problems, a Gˆateaux derivative and a Fr´echet derivative are defined as a prerequisite. The equivalence of solving a variational problem and solving a variational inequality is established. I then introduce the Sobolev space to study the solvability of differential equations. In Chapter 14, entitled ‘The Wavelet Analysis’, I provide a brief introduction to the origin of wavelet analysis. It is the outcome of the confluence of mathematics, engineering and computer science. Wavelet analysis has begun to play a serious role in a broad range of applications including signal processing, data and image compression, the solving of partial differential equations, the modeling of multiscale phenomena and statistics. Starting from the notion of information, we discuss the scalable structure of information. Next we discuss the algebra and geometry of wavelet matrices like Haar matrices and Daubechies’s matrices of different ranks. Thereafter come the one-dimensional wavelet systems where the scaling equation associated with a wavelet matrix, the expansion of a x function in terms of wavelet system associated with a matrix and other results are presented. The final chapter is concerned with dynamical systems. The theory of dynamical systems has its roots in the theory of ordinary differential equations. Henry Poincar´e and later Ivar Benedixon studied the topological properties of the solutions of autonomous ordinary differential equations (ODEs) in the plane. They did so with a view of studying the basic properties of autonomous ODEs without trying to find out the solutions of the equations. The discussion is confined to onedimensional flow only. Prerequisites The reader of the book is expected to have a knowledge of set theory, elements of linear algebra as well as having been exposed to metric spaces. Courses The book can be used to teach two semester courses at the M.Sc. level in universities (MS level in Engineering Institutes): (i) Basic course on functional analysis. For this Chapters 2–9 may be consulted. (ii) Another course may be developed on linear operator theory. For this Chapters 2, 3–5, 7–9 and 11 may be consulted. The Lebesgue measure is discussed at an elementary level in Chapter 10; Chapters 2–9 can, however, be read without any knowledge of the Lebesgue measure. Those who are interested in applications of functional analysis may look into Chapters 12 and 13. Acknowledgements I wish to express my profound gratitude to my advisor, the late Professor Parimal Kanti Ghosh, former Ghose professor in the Department of Applied Mathematics, Calcutta University, who introduced me to this subject. My indebtedness to colleagues and teachers like Professor J. G. Chakraborty, Professor S. C. Basu is duly acknowledged. Special mention must be made of my colleague and friend Professor A. Roy who constantly encouraged me to write this book. My wife Mrs. M. Sen offered all possible help and support to make this project a success, and thanks are duly accorded. I am also indebted to my sons Dr. Sugata Sen and Professor Shamik Sen for providing editorial support. Finally I express my gratitude to the inhouse editors and the external reviewer. 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