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.
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