Probabilistic normed spaces
type: Item
instance of: scholarly article
zbMATH Open document ID: 1310.46002
author: Bernardo Lafuerza Guillén
author: Panackal Harikrishnan
publication date: 2014-10-21T00:00:00Z
Mathematics Subject Classification ID: 46-02
Mathematics Subject Classification ID: 46S50
Mathematics Subject Classification ID: 54E70
zbMATH DE Number: 6359385
zbMATH Keywords: probabilistic normed space
zbMATH Keywords: \(D\)-boundedness
zbMATH Keywords: \(D\)-compactness
zbMATH Keywords: normability
zbMATH Keywords: invariant PN space
zbMATH Keywords: Menger's 2-probabilistic normed space
MaRDI profile type: Publication
full work available at URL: p944
OpenAlex ID: W2167137666
DOI: 10.1142/P944
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title: Probabilistic normed spaces
review text: The theory of probabilistic normed spaces, henceforth called PN spaces, was born as a ``natural'' consequence of the theory of probabilistic metric spaces, henceforth PM spaces. These spaces were introduced by \textit{K. Menger} [Proc. Natl. Acad. Sci. USA 28, 535--537 (1942; Zbl 0063.03886)] who introduced the idea of a statistical metric, i.e., of replacing the number \(d(p,q)\), which gives the distance between two points \(p\) and \(q\) in a non-empty set \(S\), by a distribution function \(F_{p,q}\), whose value at \(t\in [0,+\infty]\) is interpreted as the probability that the distance between the points \(p\) and \(q\) is smaller than \(t\). The theory was then brought to its present state by Schweizer and Sklar, in a series of papers. The achievements of these two authors culminated in their masterly monograph [\textit{B. Schweizer} and \textit{A. Sklar}, Probabilistic metric spaces. New York-Amsterdam-Oxford: North-Holland (1983; Zbl 0546.60010)], in which all the known results until the early 1980s were collected. This book has recently been reprinted by Dover [(2012; Zbl 1231.60008)]. The reader should read the lucid introduction by \textit{B. Schweizer} to Part 6 of \textit{K. Menger}'s [Selecta Mathematica. Vol. 2. Wien: Springer (2003; Zbl 1047.01010)], where Schweizer goes through the history of the subject. Thus PN spaces may be regarded as just a chapter, albeit an important one, in the history of PM spaces.NEWLINENEWLINEPN spaces were introduced by Serstnev in a series of papers; he was motivated by problems of best approximation in statistics. His definition runs along the same path followed in order to probabilize the notion of metric space and to introduce PM spaces. His definition turned out to be strict, perhaps too strict, so that after an initial flourishing of papers, mainly from the Russian school, the theory remained dormant for a number of years until it was revived by a new definition proposed in [\textit{C. Alsina}, \textit{B. Schweize…
reviewed by: Chuanxi Zhu
description: scientific article; zbMATH DE number 6359385