8. Modular Metric Spaces

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]