The algebraic structure of finitely generated <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msup><mml:mi>L</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="script">F</mml:mi><mml:mo>,</mml:mo><mml:mi>K</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>-modules and the Helly theorem in random normed modules

Guo, Tiexin; Shi, Guang

doi:10.1016/j.jmaa.2011.03.069

Cited by 22 publications

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“…bounded random linear functional and established the corresponding Hahn-Banach theorem (see Example 2.12), which leads to the development of random conjugate spaces. Subsequently, the notions of random normed and inner product modules were further introduced in [10,11] so that the theory of random conjugate spaces obtained a fast development, for example, the representation theory of random conjugate spaces was given in [12,24,16], the characterizations of random reflexivity under the framework of random conjugate spaces were deeply studied in [19,15], the Helly theorem in random normed modules was also established by characterizing the dimensional structure of finitely generated L 0 -modules in [20] and the geometric theory of random normed modules was deeply studied in [25]. In 1999, the notion of a random locally convex module was introduced by Guo and the theory of random conjugate spaces for random locally convex modules was also widely developed, see [11,13,14,17,26,27] and the reference literature therein.…”

Section: Introductionmentioning

confidence: 99%

Notions, Definitions, and Models

Guo

Susilo

2018

Introduction to Security Reduction

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First, we prove that a random metric space can be isometrically embedded into a complete random normed module, as an application it is easy to see that the notion of d-σ-stability in a random metric space can be regarded as a special case of the notion of σ-stability in a random normed module; as another application we give the final version of the characterization for a d-σ-stable random metric space to be stably compact. Second, we prove that an L p -normed L ∞ -module is exactly generated by a complete random normed module so that the gluing property of an L p -normed L ∞ -module can be derived from the σ-stability of the generating random normed module, as applications almost all the basic theory of module duals can be obtained from the theory of random conjugate spaces. Third, we prove that a random normed space is order complete iff it is (ε, λ)-complete, as an application it is proved that the d-decomposability of an order complete random normed space is exactly its d-σ-stability. Finally, we prove that an equivalence relation on the product space of a nonempty set X and a complete Boolean algebra B is regular iff it can be induced by a B-valued Boolean metric on X, as an application it is proved that a nonempty subset of a Boolean set (X, d) is universally complete iff it is a B-stable set defined by a regular equivalence relation.

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Section: Introductionmentioning

confidence: 99%

Notions, Definitions, and Models

Guo

Susilo

2018

Introduction to Security Reduction

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On the kernel space for an L⁰‐linear function

Zhang

Wang

Liu

2018

Math Methods in App Sciences

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A random normed module is a random generalization of an ordinary normed space, and it is the randomization that makes a random normed module possess rich stratification structures. On the basis of these stratification structures, this paper shows that either the kernel space N(f) for an L0‐linear function f from a random normed module S to the algebra L0false(scriptF,Kfalse) is a closed submodule or N(f) on some specifical stratification is a dense proper submodule of S, which generalizes the classical case. In the meantime, a characterization for the kernel space N(f) to be closed is also given.

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