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A Mazur Ulam Theorem for Probabilistic Normed Spaces Ahmad Salimi a salimi@math iut ac ir Sep 11 2011 Master of Science Thesis Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 IranSupervisor Dr Mohammad Reza Koushesh koushesh@cc iut ac ir2000 MSC Primary 46S50 Secondary 54E70 46B20 Key words Probabilistic normed spaces Mazur Ulam theorem Abstract In 1942 Karl Menger de ned statistical metric spaces by generalizing metric spaces by replacingthe distance d p q between two points p and q by a distribution function Fp q Here the value Fp q x at the point x can be interpreted as the probability that the distance between the two points p and qbe less than x Recall that a distribution function F is a real mapping de ned on the extended realline that is increasing and left continuous and such that it attains the value 0 at and 1 at Menger is original idea explained in the above resembled and pioneered the fuzzy idea proposed by L A Zadeh by several years Later Sestnev generalized statistical metric spaces de ned by Menger Further these spaces are these days known as Sestnev probabilistic normed spaces Sestnev also considered probabilistic normed spaces probabilistic normed spaces as de ned by Sestnev despitethe simplicity of their de nition had tittle program perhaps mainly due to its technical di cult thismotivated Alsina Schweizer and Sklar to reconsider the de nition of probabilistic normed spacesand coming up with a rede nition of probabilistic normed spaces A number of Authors so farhow shown interest in the new de nition and several research articles appeared so far based onthe Alsina Schweizer and Sklar s new de nition of probabilistic normed spaces The papers of dealwith topics related to topological and completeness quotient boundedness and coptteiness concepts linear operators between probabilistic normed spaces and probabilistic normes on them productand quotient of probabilistic normed spaces and xed points theorem in the probabilistic contexts A new wider de nition of a PN space was introduced in 1993 by Alsina Schweizer and Sklar Their de nition quickly became the standard one and to the best of the authors knowledge it hasbeen adopted by all the researchers who after them have investigated the properties the uses orthe applications of PN spaces Alsina Schweizer and Sklar generalized the de nition of probabilisticnormed spaces still further to de ne the probabilistic inner product spaces The latter subjectseems not to be very well researched perhaps due to its technical de nition Our results howeverhas nothing to do with the probabilistic inner product spaces as de ned by the above authors Anisometry between two normed spaces is a mapping that preservs the distance between the points i e the distance between any two points and their images are the same Also a mapping T between tworeal line spaces L1 and L2 where is called A ne if we have T p 1 q T p 1 T q for every p q L1 and R So that we will sean the de nition of isometry will correspondingwith the probabilistic normed spaces The classical form of the Mazur Ulam theorem asserts thatevery surjective isometry between normed spaces is A ne This theorem has been proved by Mazurand Ulam in 1932 D Mushtari proved in 1968 the same result in the case of probabilistic normedspaces in the sense of A Sherstnev The purpose of this thesis is to prove the theorem in the probabilistic setting as de ned by Alsina schweizer and Sklar 1

سليمي، احمد

قضيه ي مازور- اولام براي فضاهاي نرم دار احتمالي

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