A basic strict separation theorem in random locally convex modules

Guo, Tiexin; Xiao, Haifeng; Chen, Xinxiang

doi:10.1016/j.na.2009.02.038

Cited by 36 publications

(62 citation statements)

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“…The notion of a random locally convex module was first introduced in [18] and deeply developed under the (ε, λ)-topology in [19,20] for the further development of the theory of R N modules. In 2009, motivated by financial applications, Filipović, Kupper and Vogelpoth presented in [21] a new topology (called the locally L 0 -convex topology) for a random locally convex module and proved that the theory of a Hausdorff locally L 0 -convex module introduced in [21] is equivalent to the theory of a random locally convex module endowed with the locally L 0 -convex topology.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The algebraic structure of finitely generated L0(F,K)-modules and the Helly theorem in random normed modules

Guo

Shi

2011

Journal of Mathematical Analysis and Applications

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Section: Introduction and Main Resultsmentioning

confidence: 99%

The algebraic structure of finitely generated L0(F,K)-modules and the Helly theorem in random normed modules

Guo

Shi

2011

Journal of Mathematical Analysis and Applications

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“…Lemma 3.2 [21] . Let (S, {X d } d∈D ) be a random locally convex module over K with base (Ω, A, μ),p ∈ S, A ⊂ S a closed M -convex set, and p ∈ A.…”

Section: Definition 36 Let (S {X D } D∈d ) Be a Random Locally Conmentioning

confidence: 99%

Random duality

Guo

Chen

2009

Sci. China Ser. A-Math.

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The purpose of this paper is to provide a random duality theory for the further development of the theory of random conjugate spaces for random normed modules. First, the complicated stratification structure of a module over the algebra L(µ, K) frequently makes our investigations into random duality theory considerably different from the corresponding ones into classical duality theory, thus in this paper we have to first begin in overcoming several substantial obstacles to the study of stratification structure on random locally convex modules. Then, we give the representation theorem of weakly continuous canonical module homomorphisms, the theorem of existence of random Mackey structure, and the random bipolar theorem with respect to a regular random duality pair together with some important random compatible invariants.Keywords: random duality, weakly continuous canonical module homomorphism, random compatible structure, random bipolar theorem MSC(2000): 46A20, 46A16, 46H05, 46H25

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“…Further, Guo introduced the notions of random normed modules (a special class of random normed spaces) and random inner product modules in [11,12,44] (here, we also mention the work [34] of Haydon, et.al, who independently introduced the notion of random normed modules over the real number field in the name of randomly normed L 0 -modules, as a tool for the study of ultrapowers of Lebesgue-Bochner function spaces), which leads to a series of deep developments of random conjugate spaces [13,14,17,21,26] (here, we also mention the famous work [33] of Hansen and Richard, who independently proved the Riesz representation theorem of random conjugate spaces for a class of special complete random inner product modules-conditional Hilbert spaces, and gave its applications in representing the equilibrium price). As a random generalization of a locally convex space, random locally convex modules were introduced by Guo in [16] and deeply developed in [20,22,24]. It should be pointed out that random functional analysis was developed under the (ε, λ)-topology before 2009.…”

Section: Introductionmentioning

confidence: 99%

L⁰-convex compactness and its applications to random convex optimization and random variational inequalities

et al. 2020

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First, this paper introduces the notion of L 0 -convex compactness for a special class of closed convex subsets-closed L 0 -convex subsets of a Hausdorff topological module over the topological algebra L 0 (F, K), where L 0 (F, K) is the algebra of equivalence classes of random variables from a probability space (Ω, F, P ) to the scalar field K of real numbers or complex numbers, endowed with the topology of convergence in probability. Then, this paper continues to develop the theory of L 0convex compactness by establishing various kinds of characterization theorems on L 0 -convex compactness for L 0 -convex subsets of a class of important topological modules-complete random normed modules, in particular, we make full use of the theory of random conjugate spaces to establish the characterization theorem of James type on L 0 -convex compactness for a closed L 0 -convex subset of a complete random normed module, which also surprisingly implies that our notion of L 0convex compactness coincides with GordanŽitković's notion of convex compactness in the context of a closed L 0 -convex subset of a complete random normed module. As the first application of our results, we give a fundamental theorem on random convex optimization (or, L 0 -convex optimization), which includes Hansen and Richard's famous result as a special case. As the second application, we give an existence theorem of solutions of random variational inequalities, which generalizes H.Brezis' classical result from a reflexive Banach space to a random reflexive complete random normed module. It should be emphasized that a new method, namely the L 0convex compactness method, is presented for the second application since the usual weak compactness method is no longer applicable in the present case. Besides, our fundamental theorem on random convex optimization can be also applied in the study of optimization problems of conditional convex risk measures, which will be given in our future papers.

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A basic strict separation theorem in random locally convex modules

Cited by 36 publications

References 7 publications

The algebraic structure of finitely generated L0(F,K)-modules and the Helly theorem in random normed modules

The algebraic structure of finitely generated L0(F,K)-modules and the Helly theorem in random normed modules

Random duality

L⁰-convex compactness and its applications to random convex optimization and random variational inequalities

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A basic strict separation theorem in random locally convex modules

Cited by 36 publications

References 7 publications

The algebraic structure of finitely generated L0(F,K)-modules and the Helly theorem in random normed modules

The algebraic structure of finitely generated L0(F,K)-modules and the Helly theorem in random normed modules

Random duality

L0-convex compactness and its applications to random convex optimization and random variational inequalities

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L⁰-convex compactness and its applications to random convex optimization and random variational inequalities