Probabilistic normed spaces (Q2925243)

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. Schweizer} and \textit{A. Sklar}, Aequationes Math. 46, No. 1--2, 91--98 (1993; Zbl 0792.46062)], who reexamined in a profound way the definition of a classical norm, gave an equivalent formulation of a condition given by Serstnev, and proposed the new definition. In the present book, the authors deal with PN spaces according to both definitions.NEWLINENEWLINEThe whole book contains 11 chapters. The authors discuss the topology of PN spaces, probabilistic norms and convergence, products and quotients of PN spaces, \(D\)-boundedness and \(D\)-compactness, normability, invariant and semi-invariant PN spaces, stability of some functional equations in PN spaces and Menger's 2-probabilistic normed spaces. The content is self-contained and systematic. This book provides a good opportunity for scholars and students to get familiar with the theory of PN spaces and to acquire the basic knowledge in this field.