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The book presents a theory of abstract duality pairs which arises
by replacing the scalar field by an Abelian topological group in
the theory of dual pair of vector spaces. Examples of abstract
duality pairs are vector valued series, spaces of vector valued
measures, spaces of vector valued integrable functions, spaces of
linear operators and vector valued sequence spaces. These examples
give rise to numerous applications such as abstract versions of the
Orlicz-Pettis Theorem on subseries convergent series, the Uniform
Boundedness Principle, the Banach-Steinhaus Theorem, the Nikodym
Convergence theorems and the Vitali-Hahn-Saks Theorem from measure
theory and the Hahn-Schur Theorem from summability. There are no
books on the current market which cover the material in this book.
Readers will find interesting functional analysis and the many
applications to various topics in real analysis.
A functional calculus is a construction which associates with an
operator or a family of operators a homomorphism from a function
space into a subspace of continuous linear operators, i.e. a method
for defining "functions of an operator". Perhaps the most familiar
example is based on the spectral theorem for bounded self-adjoint
operators on a complex Hilbert space.This book contains an
exposition of several such functional calculi. In particular, there
is an exposition based on the spectral theorem for bounded,
self-adjoint operators, an extension to the case of several
commuting self-adjoint operators and an extension to normal
operators. The Riesz operational calculus based on the Cauchy
integral theorem from complex analysis is also described. Finally,
an exposition of a functional calculus due to H. Weyl is given.
This text is an introduction to functional analysis which requires
readers to have a minimal background in linear algebra and real
analysis at the first-year graduate level. Prerequisite knowledge
of general topology or Lebesgue integration is not required. The
book explains the principles and applications of functional
analysis and explores the development of the basic properties of
normed linear, inner product spaces and continuous linear operators
defined in these spaces. Though Lebesgue integral is not discussed,
the book offers an in-depth knowledge on the numerous applications
of the abstract results of functional analysis in differential and
integral equations, Banach limits, harmonic analysis, summability
and numerical integration. Also covered in the book are versions of
the spectral theorem for compact, symmetric operators and
continuous, self adjoint operators.
The book uses classical problems to motivate a historical
development of the integration theories of Riemann, Lebesgue,
Henstock-Kurzweil and McShane, showing how new theories of
integration were developed to solve problems that earlier
integration theories could not handle. It develops the basic
properties of each integral in detail and provides comparisons of
the different integrals. The chapters covering each integral are
essentially independent and could be used separately in teaching a
portion of an introductory real analysis course. There is a
sufficient supply of exercises to make this book useful as a
textbook.
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