Modular Metric Spaces

Mohamed A. Khamsi () and Wojciech M. Kozlowski ()
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Mohamed A. Khamsi: The University of Texas at El Paso, Department of Mathematical Sciences
Wojciech M. Kozlowski: University of New South Wales, School of Mathematics and Statistics

Chapter 8 in Fixed Point Theory in Modular Function Spaces, 2015, pp 219-234 from Springer

Abstract: Abstract The concept of a metric space is closely related to our intuitive understanding of space is the 3-dimensional Euclidean space. In fact, the notion of metric is a generalization of the Euclidean metric arising from the basic long known properties of the Euclidean distance. Maurice Fréchet1is credited as the mathematician who introduced the abstract definition of a metric space. Metric spaces are seen as a nonlinear version of vector spaces endowed with a norm. Following the same direction, one may think of a nonlinear version of modular function spaces [74]. Indeed throughout this book we have seen that a modular function space is a vector space endowed with a modular function. Therefore it is natural to consider a nonlinear version of function modular spaces. The first to consider such generalization was V. Chistyakov [46,47]. Informally speaking, whereas a metric on a set represents nonnegative finite distances between any two points of the set, a modular on a set attributes a nonnegative (possibly, infinite valued) “field of (generalized) velocities” to each “time” $\lambda> 0$ (the absolute value of) an average velocity $w_\lambda(x,y)$ which is associated in such a way that in order to cover the “distance” between points $x, y \in X$ it takes time λ to move from x to y with velocity $w_\lambda(x,y)$ . The nonlinear approach to modular function spaces was initiated in [1,2,3]

Keywords: Modular Metal; Modular Function Spaces; Nonlinear Version; Nonlinear Approach; Balanced Modulus (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/978-3-319-14051-3_8

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