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Semi-Lipschitz functions and best approximation in quasi-metric spaces. (English) Zbl 0980.41029
Let (X,d) be a quasi-metric space (symmetry of d is not satisfied). A function f:X is called semi-Lipschitz if there exists a number K0 such that fx-fyKdx,y, for all x,yX· One denotes by SLip 0 X the set of all semi-Lipschitz functions vanishing at a fixed point x 0 X· It follows that SLip 0 X is a semilinear space and the functional · d defined by f d =sup((fx-fy)0)/dx,y:x,yX,dx,y>0 is a quasi-norm on SLip 0 X· For a subset Y of X containing x 0 and pX let P Y (p)={y 0 Y:d(y 0 ,p) =inf{d(y,p):yY}}· The authors give characterizations of the elements of P Y p in terms of the elements of SLip 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 SLip 0 X is also proved. As the authors point out in the introduction, other properties of SLip 0 X (compactness, the property of being a Banach space etc.) will be studied elsewhere.
41A65Abstract approximation theory