Abstract Duality Pairs in Analysis

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.

Contents:

• Preface

• Abstract Duality Pairs or Abstract Triples

• Subseries Convergence

• Bounded Multiplier Convergent Series

• Multiplier Convergent Series

• The Uniform Boundedness Principle

• Banach–Steinhaus

• Biadditive and Bilinear Operators

• Triples with Projections

• Weak Compactness in Triples

• Appendices:

  • Topology
  • Sequence Spaces
  • Boundedness Criterion
  • Drewnowski
  • Antosik–Mikusinski Matrix Theorems

• References

• Index

Readership: Graduate Students and researchers in functional analysis.
Key Features:

• The book should be of interest to people with interests in functional analysis
• Readers should find interesting the many applications to various topics in real analysis
• There are no books on the current market which cover the material in the book