Semitopological Vector Spaces

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 Algorithms (2005), On the Nature and Essence of Mathematics (1998), Intellectual Components of Creativity (1998), Fundamental Structures of Knowledge and Information (1997), The World of Theories and Power of Mind (1992), and Axiological Aspects of Scientific Theories (1991). Dr. Burgin has also edited eight books.

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.