Semitopological Vector Spaces Hypernorms, Hyperseminorms, and Operators

This new volume shows how it is possible to further develop and essentially extend the theory of operators in infinite-dimensional vector spaces, which plays an important role in mathematics, physics, information theory, and control theory. The book describes new mathematical structures, such as hypernorms, hyperseminorms, hypermetrics, semitopological vector spaces, hypernormed vector spaces, and hyperseminormed vector spaces. It develops mathematical tools for the further development of functional analysis and broadening of its applications.

Exploration of semitopological vector spaces, hypernormed vector spaces, hyperseminormed vector spaces, and hypermetric vector spaces is the main topic of this book. A new direction in functional analysis, called quantum functional analysis, has been developed based on polinormed and multinormed vector spaces and linear algebras. At the same time, normed vector spaces and topological vector spaces play an important role in physics and in control theory.

To make this book comprehendible for the reader and more suitable for students with some basic knowledge in mathematics, denotations and definitions of the main mathematical concepts and structures used in the book are included in the appendix, making the book useful for enhancing traditional courses of calculus for undergraduates, as well as for separate courses for graduate students. The material of Semitopological Vector Spaces: Hypernorms, Hyperseminorms and Operators is closely related to what is taught at colleges and universities. It is possible to use a definite number of statements from the book as exercises for students because their proofs are not given in the book but left for the reader.

Introduction. Hypernumbers: Constructions and Operations. Hypernorms, Hypermetrics and Hyperseminorms. Hyperoperators, Hyperfunctionals and Extra functions: Constructions and Operations. From Topological Vector Spaces to Semi Topological Vector Spaces. Continuity and Boundedness. From Continuity to Fuzzy Continuity. Approximately Linear Operators. From Boundedness to Multiboundedness. Boundedness and Fuzzy Continuity. Conclusion and Directions for Future Research.

Dr. Mark Burgin received his MA and PhD in mathematics from Moscow State University, which was one of the best universities in the world at that time, and Doctor of Science in logic and philosophy from the National Academy of Sciences of Ukraine. He was a Professor at the Institute of Education, Kiev; at International Solomon University, Kiev; at Kiev State University, Ukraine; and Head of the Assessment Laboratory in the Research Center of Science at the National Academy of Sciences of Ukraine. Currently he is working at University of California, Los Angeles, USA. Dr. Burgin is a member of the New York Academy of Sciences and an Honorary Professor of the Aerospace Academy of Ukraine. Dr. Burgin is also a member of the Science Advisory Committee at Science of Information Institute, Washington. Dr. Burgin is doing research, has publications, and taught courses in various areas of mathematics, artificial intelligence, computer science, information sciences, system theory, logic, psychology, social sciences, and methodology of science. He originated such theories as system theory of time, general information theory, theory of named sets, hyperprobability theory, and neoclassical analysis (in mathematics) and has made essential contributions to such fields as foundations of mathematics, theory of algorithms and computation, theory of knowledge, theory of intellectual activity, and complexity studies. He was the first to discover non-diophantine arithmetics; the first to axiomatize and build mathematical foundations for negative probability used in physics, finance, and economics; and the first to explicitly overcome the barrier posed by the Church-Turing Thesis. Dr. Burgin has authored and co-authored more than 500 papers and 21 books, including Structural Reality (2012), Theory of Named Sets (2011), Theory of Information (2010), Neoclassical Analysis: Calculus Closer to the Real World"(2008), Super-recursive Algorithm