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On the definition of a probabilistic inner product space. (English) Zbl 0945.46012

A new definition of a probabilistic inner product space is presented which overcomes many of the difficulties encountered in earlier definitions. Instead of relying on a probabilistic generalization of the Cauchy Schwarz inequality, this approach uses a probabilistic generalization of the triangle inequality. Ordinary real inner product spaces and the EP spaces (natural generalizations of real inner product spaces to random variables) are included in this definition. Moreover, the authors show the probabilistic inner product spaces under Min are real inner product spaces. Consequently, generalizations of theorems on real inner product spaces to probabilistic inner product spaces under Min really aren’t generalizations at all.

MSC:

46C50 Generalizations of inner products (semi-inner products, partial inner products, etc.)
46S50 Functional analysis in probabilistic metric linear spaces
54E70 Probabilistic metric spaces
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