zbMATH — the first resource for mathematics

On the definition of a probabilistic normed space. (English) Zbl 0792.46062
It is given a new definition of a probabilistic normed space. Before the definition the authors somewhat change equivalently the axioms of a usual norm. The given definition includes the earlier definition of A. N. Serstnev as a special case and leads naturally to the definition of the principal class of probabilistic normed spaces, the Menger spaces.

46S50 Functional analysis in probabilistic metric linear spaces
46B09 Probabilistic methods in Banach space theory
54E99 Topological spaces with richer structures
60B11 Probability theory on linear topological spaces
Full Text: DOI EuDML
[1] Menger, K.,Statistical metrics. Proc. Nat. Acad. Sci. U.S.A.28 (1942), 535–537. · Zbl 0063.03886
[2] Schweizer, B. andSklar, A.,Probabilistic metric spaces. Elsevier North-Holland, New York, 1983.
[3] Šerstnev, A. N.,Random normed spaces: problems of completeness. Kazan. Gos. Univ. Učen. Zap.122 (1962), 3–20.
[4] Šerstnev, A. N.,On the notion of a random normed space. Dokl. Akad. Nauk SSSR149(2) (1963), 280–283.
[5] Šerstnev, A. N.,Best approximation problems in random normed spaces. Dokl. Akad. Nauk SSSR149(3) (1963), 539–542.
[6] Šerstnev, A. N.,Some best approximation problems in random normed spaces. Rev. Roumaine Math. Pures Appl.9 (1963), 771–789.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.