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Modular metric spaces. I: Basic concepts. (English) Zbl 1200.54014

The author introduces the notion of a (metric) modular as follows: A (metric) modular on a set $X$ is a function $w:\left(0,\infty \right)×X×X\to \left[0,\infty \right]$ satisfying, for all $x,y,z\in X,$ the following three properties: $x=y$ if and only if $w\left(\lambda ,x,y\right)=0$ for all $\lambda >0;$ $w\left(\lambda ,x,y\right)=w\left(\lambda ,y,x\right)$ for all $\lambda >0;$ and $w\left(\lambda +\mu ,x,y\right)\le w\left(\lambda ,x,z\right)+w\left(\mu ,y,z\right)$ for all $\lambda ,\mu >0·$

He shows that given ${x}_{0}\in X$ the set ${X}_{w}=\left\{x\in X:{lim}_{\lambda \to \infty }w\left(\lambda ,x,{x}_{0}\right)=0\right\}$ is a metric space with metric ${d}_{w}^{o}\left(x,y\right)=inf\left\{\lambda >0:w\left(\lambda ,x,y\right)\le \lambda \right\},$ which he calls a modular space. The article develops the theory of metric spaces generated by (convex) modulars. In this way the author is able to extend results from Nakano’s theory of modular linear spaces to his setting. Among other things, the method can be used to define new metric spaces of (multivalued) functions of bounded generalized variation of a real variable with values in metric semigroups and abstract convex cones.

##### MSC:
 54E35 Metric spaces, metrizability 46A80 Modular topological linear spaces