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nY%RS%hS%RSphRBSp&p%hpAhSpRSpqS!BB, 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 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. Several
improvements in form and content were made at their suggestion. xi >=C4=CB
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