A Short and Partial History of Probabilistic Normed Spaces

This contribution aims at presenting a survey of a portion of the theory of Probabilistic Normed spaces. No result will be proved, so that the reader is referred to the original sources for the proofs. The theory of PN spaces has many facets and touches on many branches of mathematics, for instance, geometry, functional analysis, topology, probability. This justifies the adjective “partial” that appears in the title. Therefore, it is perhaps better to declare from the start what one may expect from this survey. It is only natural to investigate which “classical”properties of normed spaces are preserved in the new setting. But it is probably more interesting to look for those properties that pertain to the new theory and which have no corresponding analogue in the classical theory. It must also be added that Probabilistic Normed spaces may be approached from different standpoints: they may be studied for their own sake, as a special subject in functional analysis or in topology that is worth investigating simply because it is there, or because it provides a tool to approach open problems or, again, to shed light on topics that one thought had been thoroughly investigated. In this, one thinks immediately of the possible applications in probability and statistics. This paper was written as a part of the project “Metodi stocastici in finanza matematica” of the Italian M.I.U.R..