# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Semi-Lipschitz functions and best approximation in quasi-metric spaces. (English) Zbl 0980.41029
Let $\left(X,d\right)$ be a quasi-metric space (symmetry of $d$ is not satisfied). A function $f:X\to ℝ$ is called semi-Lipschitz if there exists a number $K\ge 0$ such that $f\left(x\right)-f\left(y\right)\le Kd\left(x,y\right)$, for all $x,y\in X·$ One denotes by $SLi{p}_{0}X$ the set of all semi-Lipschitz functions vanishing at a fixed point ${x}_{0}\in X·$ It follows that $SLi{p}_{0}X$ is a semilinear space and the functional ${∥·∥}_{d}$ defined by ${∥f∥}_{d}=sup\left\{\left(\left(f\left(x\right)-f\left(y\right)\right)\vee 0\right)/d\left(x,y\right)\right\:x,y\in X,d\left(x,y\right)>0}$ is a quasi-norm on $SLi{p}_{0}X·$ For a subset $Y$ of $X$ containing ${x}_{0}$ and $p\in X$ let ${P}_{Y}\left(p\right)=\left\{{y}_{0}\in Y:d\left({y}_{0},p\right)$ $=inf\left\{d\left(y,p\right):y\in Y\right\}\right\}·$ The authors give characterizations of the elements of ${P}_{Y}\left(p\right)$ in terms of the elements of $SLi{p}_{0}X·$ One obtains results similar to those obtained in the case of metric spaces and the spaces of Lipschitz functions on them (which in their turn are inspired by the characterizations of the elements of best approximation in normed spaces in terms of the elements of their duals). The completeness of the space $SLi{p}_{0}X$ is also proved. As the authors point out in the introduction, other properties of $SLi{p}_{0}X$ (compactness, the property of being a Banach space etc.) will be studied elsewhere.
##### MSC:
 41A65 Abstract approximation theory
##### Keywords:
quasi-metric space; best approximation