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Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Functional analysis
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Elementary Operator Theory (Paperback)
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Elementary Operator Theory (Paperback)
Series: De Gruyter Textbook
Expected to ship within 18 - 22 working days
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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
General
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