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Статья

Modular metric spaces, I: Basic concepts

Nonlinear Analysis. 2010. Vol. 72. No. 1. P. 1-14.

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,zX, 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={xX:limλw(λ,x,x0)=0} is a metric space with metric