Semi-Lipschitz Functions and Best Approximation in Quasi-Metric Spaces

By Salvador Romaguera, M. Sanchis. 2000

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Duality and quasi-normability for complexity spaces (2002, background): “We start this section giving the definitions of a quasi-norm and of a quasinormed space in the sense of [5], [6] and [21] (see [4] for the related notion of a nonsymmetric norm).”
The Dual Space of an Asymmetric Normed Linear Space (2003, methodology): “Asymmetric normed (semi)linear spaces and other related structures provide suitable tools in some fields of Theoretical Computer Science and Approximation Theory, respectively (see [16], [17], [14], [12], etc.”
Functional Analysis in Asymmetric Normed Spaces (2012, background): “functions, one deﬁnes an extended quasi-metric on SLip(X) by the formula (7.8) δρ(f,g) = sup u((f−g)(x) −(f−g)(y)) ρ(x,y) : x,y∈ X,ρ(x,y) >0 . The following example, given by Romaguera and Sanchis [149], shows that δρ could be eﬀectively an extended quasi-metric. Example 7.8. For x,y∈ R let ρ(x,y”