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October 12, 2026
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?

О свойствах модулярных пространств

С. 394–398.
В.В.Чистяков

We present basic concepts of the theory of modular spaces on arbitrary sets, which

extends simultaneously the theory of such spaces on linear sets and the theory of

metric spaces. We study the relationship between (three) modular spaces and metrics

on them in the convex and nonconvex cases. We define the modular notions of

convergence, topology and completeness. We exhibit transformations of modular

spaces (for the right inverse modular), connected with their duality. The assertions

are illustrated by examples.

 

Language: Russian
Full text
Keywords: модулярная сходимостьmodular convergencemodular spacemodular completenessмодулярное пространствомодулярная полнотаmodularright inversedelta-2 conditionмодуляраправая обратнаядельта-2 условие
Publication based on the results of:
Модулярные пространства, являющиеся модулярно полными (Modular spaces that are modular complete) (2017)

In book

Труды Математического центра имени Н.И.Лобачевского
Т. 54: Теория функций, ее приложения и смежные вопросы. , Каз.: Издательство Казанского математического общества и Академии наук РТ, 2017.
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В.В.Чистяков, В кн.: Труды Математического центра имени Н.И.Лобачевского. Т.60 // Материалы Международной конференции по алгебре, анализу и геометрии 2021.: Каз.: Издательство Академии наук Республики Татарстан, 2021. С. 330–332.
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Added: August 27, 2021
Modular functional spaces
Vyacheslav V. Chistyakov, , in: International Conference on Geometric Analysis in honor of the 90th anniversary of academician Yu.G.Reshetnyak.: Novosibirsk: PPC NSU, 2019. P. 40–43.
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Added: October 2, 2019
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The notion of a metric modular on an arbitrary set and the corresponding modular spaces, generalizing classical modulars over linear spaces and Orlicz spaces, were recently introduced and studied by the author [Chistyakov: Dokl. Math. 73(1):32–35, 2006 and Nonlinear Anal. 72(1):1–30, 2010]. In this chapter we present yet one more application of the metric modulars ...
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The notion of a (metric) modular on an arbitrary set and the corresponding modular space, more general than a metric space, were introduced and studied recently by the author [V.V. Chistyakov, Metric modulars and their application, Dokl. Math. 73 (1) (2006) 32–35, and Modular metric spaces, I: Basic concepts, Nonlinear Anal. 72 (1) (2010) 1–14]. ...
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The notion of a modular is introduced as follows. A (metric) modular on a set X is a function w:(0,∞)×X×X→[0,∞] satisfying, for all x,y,z∈X, the following three properties: x=y if and only if w(λ,x,y)=0 for all λ>0; w(λ,x,y)=w(λ,y,x) for all λ>0; w(λ+μ,x,y)≤w(λ,x,z)+w(μ,y,z) for all λ,μ>0. We show that, given x0∈X, the set Xw={x∈X:limλ→∞w(λ,x,x0)=0} is a ...
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Added: September 26, 2012
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Added: September 7, 2012
Неподвижные точки модулярно сжимающих отображений
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In the framework of modular metric spaces, introduced by the author in 2006, we define a new notion of modular convergence, which is more weak than the metric convergence, and establish the necessary and sufficient condition on the modular under consideration, under which the modular convergence is equivalent to the metric one. We introduce the notion of modular contractive maps, ...
Added: September 5, 2012
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