Modular metric spaces, I: Basic concepts |
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Authors: | Vyacheslav V Chistyakov |
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Institution: | Department of Applied Mathematics and Informatics, State University Higher School of Economics, Bol’shaya Pechërskaya Street 25/12, Nizhny Novgorod 603155, Russia |
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Abstract: | 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 metric space with metric , called a modular space. The modular w is said to be convex if (λ,x,y)?λw(λ,x,y) is also a modular on X. In this case Xw coincides with the set of all x∈X such that w(λ,x,x0)<∞ for some λ=λ(x)>0 and is metrizable by . Moreover, if or , then ; otherwise, the reverse inequalities hold. We develop the theory of metric spaces, generated by modulars, and extend the results by H. Nakano, J. Musielak, W. Orlicz, Ph. Turpin and others for modulars on linear spaces. |
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Keywords: | 46A80 54E35 |
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