Introduction

This introduction presents an overview of the key concepts discussed in the subsequent chapters of this book. The book presents study of approximately linear operators and their fuzzy properties, including fuzzy continuity, approximate linearity and multiboundedness, provides potent tools for applications in theoretical physics, as well as in information theory and practice. It describes extensions of number systems to hypernumbers and function spaces to extrafunction spaces. The book describes to understand that real hypernumbers differ from other generalizations of real numbers such as hyperreal numbers from nonstandard analysis introduced by Robinson, number systems with numerical sequences as infinitesimals introduced by Chwistek and studied by Schmieden and Laugwitz (1958). It constructs Hyperfunctionals and hyperoperators that extend extrafunctions as their particular case. The book studies relations between relative continuity and relative boundedness as essential properties of operators in polyhyperseminormed and polyhypernormed vector spaces.