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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.

MSC:
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
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References:
[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.
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