Continuity Properties of Probabilistic Norms

✍ Scribed by Claudi Alsina; Berthold Schweizer; Abe Sklar

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 165 KB
• Volume: 208
• Category: Article
• ISSN: 0022-247X
• DOI: 10.1006/jmaa.1997.5333

✦ Synopsis

Probabilistic normed PN spaces are real linear spaces in which the norm of each vector is an appropriate probability distribution function řather than a number. Such spaces were first introduced by A. N. Serstnev w x in 1963 3 . w x In a recent paper 1 , we gave a new definition of PN spaces that ǐncludes Serstnev's and leads naturally to the identification of the principal class of PN spaces, the Menger spaces. In the present paper, we investigate questions of continuity in PN spaces under the new definition.

Just as the norm in an ordinary normed space induces a topology on the space, so the probabilistic norm in a PN space induces a natural topology on the space with respect to which this norm is continuous. It then follows readily that the operation of vector addition is continuous with respect to the norm-induced topology. Similarly, multiplication by a fixed scalar behaves as expected, though in contrast to the situation in ordinary U The author was supported by ONR Contract N-00014-87-K-0379.