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Some basic theories of random normed linear spaces and random inner product spaces. (English) Zbl 0965.46010
This paper begins by giving another formulation of the original definition of a random metric space. The present formulation is not only equivalent to the original one but also makes a random metric space automatically fall into the framework of a generalized metric space, and some new problems on topological structures are also considered. Motivated by the formulation of a random metric space mentioned above, this paper, then, explicitly presents a corresponding form of the definition of a random normed space and simplifies the definition of a random normed module. Meanwhile this paper also shows that the quotient space of an $E$-normed space is isomorphically isometric to a canonical $E$-normed space. Further, under the framework of probabilistic pseudo-normed spaces, this paper shows a probabilistic pseudo-normed linear space is a pseudo-inner product generated space iff it is isomorphically isometric to an $E$-inner product space. [This result answers an open problem recently presented by {\it C. Alsina, B. Schweizer} et al., Rend. Mat. Appl., VII. Ser. 7, No. 1, 115-127 (1997; Zbl 0945.46012).]. Finally, based on the preceding preliminaries, this paper turns to its central part: the investigations on basic theories of random inner product spaces and random inner product modules. In this part, this paper gives a deep discussion of interesting and complicated orthogonality problems. This also further motivates us to generalize G. Stampacchia’s general projection theorem from real Hilbert spaces, in appropriate form, to real complete random inner product modules.

46C50Generalizations of inner products
46S50Functional analysis in probabilistic metric linear spaces
46B09Probabilistic methods in Banach space theory
54E70Probabilistic metric spaces
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
54H25Fixed-point and coincidence theorems in topological spaces