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The book assists Calculus students to gain a better understanding
and command of integration and its applications. It reaches to
students in more advanced courses such as Multivariable Calculus,
Differential Equations, and Analysis, where the ability to
effectively integrate is essential for their success.Keeping the
reader constantly focused on the three principal epistemological
questions: 'What for?', 'Why?', and 'How?', the book is designated
as a supplementary instructional tool and consists ofThe Answers to
all the 192 Problems are provided in the Answer Key. The book will
benefit undergraduates, advanced undergraduates, and members of the
public with an interest in science and technology, helping them to
master techniques of integration at the level expected in a
calculus course.
The book assists Calculus students to gain a better understanding
and command of integration and its applications. It reaches to
students in more advanced courses such as Multivariable Calculus,
Differential Equations, and Analysis, where the ability to
effectively integrate is essential for their success.Keeping the
reader constantly focused on the three principal epistemological
questions: 'What for?', 'Why?', and 'How?', the book is designated
as a supplementary instructional tool and consists ofThe Answers to
all the 192 Problems are provided in the Answer Key. The book will
benefit undergraduates, advanced undergraduates, and members of the
public with an interest in science and technology, helping them to
master techniques of integration at the level expected in a
calculus course.
The book is intended as a text for a one-semester graduate course
in operator theory to be taught "from scratch'', not as a sequel to
a functional analysis course, with the basics of the spectral
theory of linear operators taking the center stage. The book
consists of six chapters and appendix, with the material flowing
from the fundamentals of abstract spaces (metric, vector, normed
vector, and inner product), the Banach Fixed-Point Theorem and its
applications, such as Picard's Existence and Uniqueness Theorem,
through the basics of linear operators, two of the three
fundamental principles (the Uniform Boundedness Principle and the
Open Mapping Theorem and its equivalents: the Inverse Mapping and
Closed Graph Theorems), to the elements of the spectral theory,
including Gelfand's Spectral Radius Theorem and the Spectral
Theorem for Compact Self-Adjoint Operators, and its applications,
such as the celebrated Lyapunov Stability Theorem. Conceived as a
text to be used in a classroom, the book constantly calls for the
student's actively mastering the knowledge of the subject matter.
There are problems at the end of each chapter, starting with
Chapter 2 and totaling at 150. Many important statements are given
as problems and frequently referred to in the main body. There are
also 432 Exercises throughout the text, including Chapter 1 and the
Appendix, which require of the student to prove or verify a
statement or an example, fill in certain details in a proof, or
provide an intermediate step or a counterexample. They are also an
inherent part of the material. More difficult problems are marked
with an asterisk, many problems and exercises are supplied with
"existential'' hints. The book is generous on Examples and contains
numerous Remarks accompanying definitions, examples, and statements
to discuss certain subtleties, raise questions on whether the
converse assertions are true, whenever appropriate, or whether the
conditions are essential. With carefully chosen material, proper
attention given to applications, and plenty of examples, problems,
and exercises, this well-designed text is ideal for a one-semester
Master's level graduate course in operator theory with emphasis on
spectral theory for students majoring in mathematics, physics,
computer science, and engineering. Contents Preface Preliminaries
Metric Spaces Vector Spaces, Normed Vector Spaces, and Banach
Spaces Linear Operators Elements of Spectral Theory in a Banach
Space Setting Elements of Spectral Theory in a Hilbert Space
Setting Appendix: The Axiom of Choice and Equivalents Bibliography
Index
The philosophy of the book, which makes it quite distinct from many
existing texts on the subject, is based on treating the concepts of
measure and integration starting with the most general abstract
setting and then introducing and studying the Lebesgue measure and
integration on the real line as an important particular case. The
book consists of nine chapters and appendix, with the material
flowing from the basic set classes, through measures, outer
measures and the general procedure of measure extension, through
measurable functions and various types of convergence of sequences
of such based on the idea of measure, to the fundamentals of the
abstract Lebesgue integration, the basic limit theorems, and the
comparison of the Lebesgue and Riemann integrals. Also, studied are
Lp spaces, the basics of normed vector spaces, and signed measures.
The novel approach based on the Lebesgue measure and integration
theory is applied to develop a better understanding of
differentiation and extend the classical total change formula
linking differentiation with integration to a substantially wider
class of functions. Being designed as a text to be used in a
classroom, the book constantly calls for the student's actively
mastering the knowledge of the subject matter. There are problems
at the end of each chapter, starting with Chapter 2 and totaling at
125. Many important statements are given as problems and frequently
referred to in the main body. There are also 358 Exercises
throughout the text, including Chapter 1 and the Appendix, which
require of the student to prove or verify a statement or an
example, fill in certain details in a proof, or provide an
intermediate step or a counterexample. They are also an inherent
part of the material. More difficult problems are marked with an
asterisk, many problems and exercises are supplied with
``existential'' hints. The book is generous on Examples and
contains numerous Remarks accompanying definitions, examples, and
statements to discuss certain subtleties, raise questions on
whether the converse assertions are true, whenever appropriate, or
whether the conditions are essential. With plenty of examples,
problems, and exercises, this well-designed text is ideal for a
one-semester Master's level graduate course on real analysis with
emphasis on the measure and integration theory for students
majoring in mathematics, physics, computer science, and
engineering. A concise but profound and detailed presentation of
the basics of real analysis with emphasis on the measure and
integration theory. Designed for a one-semester graduate course,
with plethora of examples, problems, and exercises. Is of interest
to students and instructors in mathematics, physics, computer
science, and engineering. Prepares the students for more advanced
courses in functional analysis and operator theory. Contents
Preliminaries Basic Set Classes Measures Extension of Measures
Measurable Functions Abstract Lebesgue Integral Lp Spaces
Differentiation and Integration Signed Measures The Axiom of Choice
and Equivalents
While there is a plethora of excellent, but mostly "tell-it-all''
books on the subject, this one is intended to take a unique place
in what today seems to be a still wide open niche for an
introductory text on the basics of functional analysis to be taught
within the existing constraints of the standard, for the United
States, one-semester graduate curriculum (fifteen weeks with two
seventy-five-minute lectures per week). The book consists of seven
chapters and an appendix taking the reader from the fundamentals of
abstract spaces (metric, vector, normed vector, and inner product),
through the basics of linear operators and functionals, the three
fundamental principles (the Hahn-Banach Theorem, the Uniform
Boundedness Principle, the Open Mapping Theorem and its
equivalents: the Inverse Mapping and Closed Graph Theorems) with
their numerous profound implications and certain interesting
applications, to the elements of the duality and reflexivity
theory. Chapter 1 outlines some necessary preliminaries, while the
Appendix gives a concise discourse on the celebrated Axiom of
Choice, its equivalents (the Hausdorff Maximal Principle, Zorn's
Lemma, and Zermello's Well-Ordering Principle), and ordered sets.
Being designed as a text to be used in a classroom, the book
constantly calls for the student's actively mastering the knowledge
of the subject matter. It contains 112 Problems, which are
indispensable for understanding and moving forward. Many important
statements are given as problems, a lot of these are frequently
referred to and used in the main body. There are also 376 Exercises
throughout the text, including Chapter 1 and the Appendix, which
require of the student to prove or verify a statement or an
example, fill in necessary details in a proof, or provide an
intermediate step or a counterexample. They are also an inherent
part of the material. More difficult problems are marked with an
asterisk, many problem and exercises being supplied with
"existential'' hints. The book is generous on Examples and contains
numerous Remarks accompanying every definition and virtually each
statement to discuss certain subtleties, raise questions on whether
the converse assertions are true, whenever appropriate, or whether
the conditions are essential. The prerequisites are set
intentionally quite low, the students not being assumed to have
taken graduate courses in real or complex analysis and general
topology, to make the course accessible and attractive to a wider
audience of STEM (science, technology, engineering, and
mathematics) graduate students or advanced undergraduates with a
solid background in calculus and linear algebra. With proper
attention given to applications, plenty of examples, problems, and
exercises, this well-designed text is ideal for a one-semester
graduate course on the fundamentals of functional analysis for
students in mathematics, physics, computer science, and
engineering. Contents Preliminaries Metric Spaces Normed Vector and
Banach Spaces Inner Product and Hilbert Spaces Linear Operators and
Functionals Three Fundamental Principles of Linear Functional
Analysis Duality and Reflexivity The Axiom of Choice and
Equivalents
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