A First Course in Functional Analysis
486 pages
English

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486 pages
English
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Description

A comprehensive introduction to functional analysis, starting from the fundamentals and extending into theory and applications across multiple disciplines.


‘A First Course in Functional Analysis: Theory and Applications’ provides a comprehensive introduction to functional analysis, beginning with the fundamentals and extending into theory and applications. The volume starts with an introduction to sets and metric spaces and the notions of convergence, completeness and compactness, and continues to a detailed treatment of normed linear spaces and Hilbert spaces. The reader is then introduced to linear operators and functionals, the Hahn-Banach theorem on linear bounded functionals, conjugate spaces and adjoint operators, and the space of linear bounded functionals. Further topics include the closed graph theorem, the open mapping theorem, linear operator theory including unbounded operators, spectral theory, and a brief introduction to the Lebesgue measure. The cornerstone of the book lies in the motivation for the development of these theories, and applications that illustrate the theories in action.


One of the many strengths of this book is its detailed discussion of the theory of compact linear operators and their relationship to singular operators. Applications in optimal control theory, variational problems, wavelet analysis and dynamical systems are highlighted.


This volume strikes an ideal balance between concision of mathematical exposition and offering complete explanatory materials and careful step-by-step instructions. It will serve as a ready reference not only for students of mathematics, but also students of physics, applied mathematics, statistics and engineering.One of the many strengths of the book is the detailed discussion of the theory of compact linear operators and their relationship to singular operators. Applications in optimal control theory, variational problems, wavelet analysis, and dynamical systems are highlighted.


This volume strikes the ideal balance between concision of mathematical exposition, and complete explanatory material accompanied by careful step-by-step instructions intended to serve as a ready reference not only for students of mathematics, but also students of physics, applied mathematics, statistics and engineering.


Introduction; I. Preliminaries; II. Normed Linear Spaces; III. Hilbert Space; IV. Linear Operators; V. Linear Functionals; VI. Space of Bounded Linear Functionals; VII. Closed Graph Theorem and Its Consequences; VIII. Compact Operators on Normed Linear Spaces; IX. Elements of Spectral Theory of Self-Adjoint Operators in Hilbert Spaces; X. Measure and Integration Lp Spaces; XI. Unbounded Linear Operators; XII. The Hahn-Banach Theorem and Optimization Problems; XIII. Variational Problems; XIV. The Wavelet Analysis; XV. Dynamical Systems; List of Symbols; Bibliography; Index

Sujets

Informations

Publié par
Date de parution 01 février 2013
Nombre de lectures 0
EAN13 9780857282224
Langue English
Poids de l'ouvrage 2 Mo

Informations légales : prix de location à la page 0,0080€. Cette information est donnée uniquement à titre indicatif conformément à la législation en vigueur.

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Preface
This book is the outgrowth of the lectures delivered on functional
analysis and allied topics to the postgraduate classes in the Department
of Applied Mathematics, Calcutta University, India. 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 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
metricstructureitiscalledametriclinearspace. Anormedlinearspaceisa
type of metric linear space, and for every element x of the space there exists
a positive number called norm x orx fulfilling certain axioms. A normed
linear space can always be reduced to a metric space by the choice of a
suitablemetric. Ideasofconvergenceinnormandcompletenessofanormed
linear space are introduced with examples of several normed linear spaces,
Banach spaces (complete normed linear spaces) and incomplete normed
linear spaces.
viiContinuity 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,
orthogonalityofvectors, theCauchy-Bunyakovsky-Schwartzinequality, 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 op 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 int
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 L ([a,b])p
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
closedgraphtheorem, whichestablishestheconditionsunderwhichaclosed
linear operator is bounded, is provided. After introducing the concept of an
viiiopenmapping,theopenmappingtheoremandtheboundedinversetheorem
are proved. Application of the open mapping theorem is also provided. The
next chapterbears 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 alinearoperatorarestudied. ThenotionoftheFredholmalternative
is discussed and the relevant theorems are provi

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