2008
DOI: 10.1007/s11425-008-0047-6
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The relation of Banach-Alaoglu theorem and Banach-Bourbaki-Kakutani-Šmulian theorem in complete random normed modules to stratification structure

Abstract: Let (Ω, A, μ) be a probability space, K the scalar field R of real numbers or C of complex numbers,and (S, X ) a random normed space over K with base (Ω, A, μ). Denote the support of (S, X ) by E, namely E is the essential supremum of the set {A ∈ A : there exists an element p in S such that Xp(ω) > 0 for almost all ω in A}. In this paper, Banach-Alaoglu theorem in a random normed space is first established as follows: The random closed unit ball S * (1) = {f ∈ S * : X * f 1} of the random conjugate space (S *… Show more

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Cited by 48 publications
(59 citation statements)
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“…Based on the new version of an R N space we presented a definitive definition of the random conjugate space for an R N space, further the deep development of the theory of random conjugate spaces led us to present the notion of a random normed module (briefly, an R N module) in [10], which is the elaboration of the notion of the original R N module introduced in [11]. With the notions of R N modules and their random conjugate spaces at hand, we have developed deeply and systematically the theory of R N modules under the (ε, λ)-topology [12][13][14][15][16]. An interesting phenomenon is: some classical theorems such as the Riesz's representation theorem in Hilbert spaces and the James theorem in Banach spaces still hold in complete random inner product modules (briefly, R I P modules) and complete R N modules, respectively [13,15], whereas others such as the classical Banach-Alaoglu theorem and Banach-Bourbaki-Kakutani-Šmulian theorem do not universally hold in our random setting [16].…”
Section: Introduction and Main Resultsmentioning
confidence: 99%
“…Based on the new version of an R N space we presented a definitive definition of the random conjugate space for an R N space, further the deep development of the theory of random conjugate spaces led us to present the notion of a random normed module (briefly, an R N module) in [10], which is the elaboration of the notion of the original R N module introduced in [11]. With the notions of R N modules and their random conjugate spaces at hand, we have developed deeply and systematically the theory of R N modules under the (ε, λ)-topology [12][13][14][15][16]. An interesting phenomenon is: some classical theorems such as the Riesz's representation theorem in Hilbert spaces and the James theorem in Banach spaces still hold in complete random inner product modules (briefly, R I P modules) and complete R N modules, respectively [13,15], whereas others such as the classical Banach-Alaoglu theorem and Banach-Bourbaki-Kakutani-Šmulian theorem do not universally hold in our random setting [16].…”
Section: Introduction and Main Resultsmentioning
confidence: 99%
“…to a weak*-weak* topology on E , we get a module analogy to the Alaoglu-Bourbaki theorem in Section 12. Theorems of this kind can also be found in [20]. Finally, we give in Section 13 a module version of Goldstine's theorem and a characterization of reflexive Banach λ-modules.…”
Section: Introductionmentioning
confidence: 99%
“…In the last ten years, the theory of RN modules together with their random conjugate spaces have obtained systematic and deep developments [8,9,[11][12][13][14][15][16][17][18][19][20]. In particular, the recently developed L 0 -convex analysis, which has been a powerful tool for the study of conditional risk measures, is just based on the theory of RN modules together with their random conjugate spaces [5,12,13].…”
Section: Introductionmentioning
confidence: 99%