A study on probabilistic normed space

Abstract

Nature defies deterministic study. So Mathematicians attempted newlineto develop a theory which obeys the Nature s behaviours. One among newlinethem is Menger, who in 1942 introduced statistical Metric which is de- newlineviated from deterministicMathematical concepts. Subsequently Fuzzy newlinetheory flourished. L. A. Zadeh introduced Fuzzy Set theory in 1965 newlinedescribing Fuzziness Mathematically for the first time. Since then, newlinea considerable body of literature has blossomed around the concept newlineof Fuzzy Sets. In fact Fuzzy Set extended the basic Mathematical newlineconcepts. Fuzzy set theory is a thrust area and give scope for Mathe- newlinematicians to contribute more on future research. We attempt to study newlineprobabilistic normed space. The purpose of this research is to study newlinethe basic concepts in probabilistic norm and to study the behaviours newlineof probabilistic normed space. The Topology induced by the prob- newlineabilistic norm and its characters are also studied. We prove closed newlinegraph theorem and some concepts on Probabilistic Banach spaces are newlineestablished. Also we estabilish a connection between a compact set newlineand finite dimensional probabilistic normed space. Further we intro- newlineduce the notion of t-best approximation, F-best approximation and newlineprove some interesting theorems. Finally we prove some fixed point newlinetheorems for contraction type mapping in probabilistic normed space. newlineThis research work carries hope for future development on probabilis- newlinetic normed space newline