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Continuity properties of probabilistic norms. (English) Zbl 0903.46075
A. N. Serstnev introduced the notion of probabilistic spaces which was recently modified by C. Alsina, B. Schweizer and A. Sklar [Aequationes Math. 46, No. 1-2, 91-98 (1993; Zbl 0792.46062)]. In this interesting paper, the authors investigate questions of continuity in probabilistic normed spaces. It is proved that vector addition is continuous with respect to the norm-induced topology where as multiplication by a fixed scalar can be discontinuous.
Reviewer: Ismat Beg (Kuwait)

MSC:
46S50 Functional analysis in probabilistic metric linear spaces
46B09 Probabilistic methods in Banach space theory
54E70 Probabilistic metric spaces
60B11 Probability theory on linear topological spaces
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[1] Alsina, C.; Schweizer, B.; Sklar, A., On the definition of a probabilistic normed space, Aequationes math., 46, 91-98, (1993) · Zbl 0792.46062
[2] Schweizer, B.; Sklar, A., Probabilistic metric spaces, (1983), Elsevier/North-Holland New York · Zbl 0546.60010
[3] ҆erstnev, A.N., On the notion of a random normed space, Dokl. akad. nauk SSSR, 149, 280-283, (1963)
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