## 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 $$f:X\rightarrow \mathbb{R}$$ is called semi-Lipschitz if there exists a number $$K\geq 0$$ such that $$f\left( x\right) -f\left( y\right) \leq Kd\left( x,y\right)$$, for all $$x,y\in X.$$ One denotes by $$SLip_0X$$ the set of all semi-Lipschitz functions vanishing at a fixed point $$x_0\in X.$$ It follows that $$SLip_0X$$ is a semilinear space and the functional $$\left\|\cdot \right\|_d$$ defined by $$\left\|f\right\|_d=\sup \left\{ ((f\left( x\right) -f\left( y\right))\vee 0)/d\left( x,y\right) \right. :\left. x,y\in X,d\left( x,y\right) >0\right\}$$ is a quasi-norm on $$SLip_0X.$$ For a subset $$Y$$ of $$X$$ containing $$x_0$$ and $$p\in X$$ let $$P_Y(p)=\{y_0\in Y:d(y_0,p)$$ $$=\inf \{d(y,p):y\in Y\}\}.$$ The authors give characterizations of the elements of $$P_Y\left( p\right)$$ in terms of the elements of $$SLip_0X.$$ 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 $$SLip_0X$$ is also proved. As the authors point out in the introduction, other properties of $$SLip_0X$$ (compactness, the property of being a Banach space etc.) will be studied elsewhere.

### MSC:

 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)

### Keywords:

quasi-metric space; best approximation
Full Text:

### References:

 [1] Deák, J., A bitopological view of quasi-uniform completeness, I, Studia Sci. Math. Hungar., 30, 389-409 (1995) · Zbl 0907.54019 [2] Doitchinov, D., On completeness in quasi-metric spaces, Topology Appl., 30, 127-148 (1988) · Zbl 0668.54019 [3] Ferrer, J.; Gregori, V.; Alegre, C., Quasi-uniform structures in linear lattices, Rocky Mountain J. Math., 23, 877-884 (1993) · Zbl 0803.46007 [4] Fletcher, P.; Hunsaker, W., Completeness using pairs of filters, Topology Appl., 44, 149-155 (1992) · Zbl 0770.54027 [5] Fletcher, P.; Lindgren, W. F., Quasi-Uniform Spaces (1982), Dekker: Dekker New York · Zbl 0402.54024 [6] Hicks, T. L., Fixed point theorems for quasi-metric spaces, Math. Japon., 33, 231-236 (1988) · Zbl 0642.54047 [7] Jachymski, J., A contribution to fixed point theory in quasi-metric spaces, Publ. Math. Debrecen, 43, 283-288 (1993) · Zbl 0814.47061 [8] Johnson, J. A., Banach spaces of Lipschitz functions and vector-valued Lipschitz functions, Trans. Amer. Math. Soc., 148, 147-169 (1970) · Zbl 0194.43603 [9] Khalimsky, E. D.; Kopperman, R. D.; Meyer, P. R., Computer graphics and connected topologies on finite ordered sets, Topology Appl., 36, 1-17 (1990) · Zbl 0709.54017 [10] Kong, T. Y.; Kopperman, R. D.; Meyer, P. R., A topological approach to digital topology, Amer. Math. Monthly, 98, 901-917 (1991) · Zbl 0761.54036 [11] Kopperman, R. D., All topologies come from generalized metrics, Amer. Math. Monthly, 95, 89-97 (1988) · Zbl 0653.54020 [12] Kovalevsky, V.; Kopperman, R. D., Some topology-based image processing algorithms, (Andina, S.; Itzkowitz, G.; Kong, T. Y.; Kopperman, R.; Misra, P. R.; Narici, L.; Told, A., Papers on General Topology and Applications. Papers on General Topology and Applications, Ann. New York Acad. Sci. (1994), New York Acad. Sci: New York Acad. Sci New York), 174-182 · Zbl 0913.68216 [13] Künzi, H. P.A., On quasi-uniform convergence and quiet spaces, Questions Answers Gen. Topology, 13, 87-92 (1995) · Zbl 0818.54014 [14] Künzi, H. P.A., Nonsymmetric topology, Topology with Applications. Topology with Applications, Bolyai Soc. Math. Studies, 4 (1993), Szekszard, p. 303-338 · Zbl 0888.54029 [15] Künzi, H. P.A.; Ryser, C., The Bourbaki quasi-uniformity, Topology Proc., 20, 161-183 (1995) · Zbl 0876.54022 [16] Matthews, S. G., Partial metrics, (Andina, S.; Itzkowitz, G.; Kong, T. Y.; Kopperman, R.; Misra, P. R.; Narici, L.; Told, A., Papers on General Topology and Applications. Papers on General Topology and Applications, Ann. New York Acad. Sci. (1994), New York Acad. Sci: New York Acad. Sci New York), 183-197 · Zbl 0911.54025 [17] Mustata, C., On the best approximation in metric spaces, Ann. Numer. Theory Approx., 4, 45-50 (1975) · Zbl 0353.41012 [18] Mustata, C., A characterization of semi-Chebyschev sets in a metric space, Ann. Numer. Theory Approx., 7, 169-170 (1978) · Zbl 0409.41010 [19] Narang, T. D., On some approximation problems in metrics spaces, Tamkang J. Math., 22, 99-103 (1991) · Zbl 0717.41047 [20] Papadopoulos, B. K., Ascoli’s theorem in a quiet quasi-uniform space, Panamer. Math. J., 3, 19-22 (1993) · Zbl 0844.54011 [21] Papadopoulos, B. K., A note on the paper “Quasi-uniform convergence on function spaces”, Questions Answers Gen. Topology, 13, 55-56 (1995) · Zbl 0818.54016 [22] Pasquale, A., Hypertopologies induced by scales of functional kind, Boll. Un. Mat. Ital. B, 7, 431-449 (1993) · Zbl 0852.54011 [23] H. Render, Generalized uniform spaces and applications to function spaces, preprint.; H. Render, Generalized uniform spaces and applications to function spaces, preprint. · Zbl 0909.54023 [24] Romaguera, S., Left K-completeness in quasi-metric spaces, Math. Nachr., 157, 15-23 (1992) · Zbl 0784.54027 [25] Romaguera, S.; Checa, E., Continuity of contractive mappings in complete quasi-metric spaces, Math. Japon., 35, 137-139 (1990) · Zbl 0705.54035 [26] Romaguera, S.; Ruiz-Gómez, M., Bitopologies and quasi-uniformities on spaces of continuous functions, I, Publ. Math. Debrecen, 47, 81-93 (1995) · Zbl 0851.54030 [27] Schellekens, M., The Smyth completion: A common foundation for denotational semantics and complexity analysis, Electron. Notes Comput. Sci., 1, 1-22 (1995) · Zbl 0910.68135 [28] M. B. Smyth, Completion of quasi-uniform spaces in terms of filters, Department of Computing, Imperial College, 1987.; M. B. Smyth, Completion of quasi-uniform spaces in terms of filters, Department of Computing, Imperial College, 1987. [29] Smyth, M. B., Quasi-uniformities: Reconciling domains with metric spaces, Third Workshop on Mathematical Foundations of Programming Language Semantics. Third Workshop on Mathematical Foundations of Programming Language Semantics, Lecture Notes in Computer Science, 289 (1987), Springer-Verlag: Springer-Verlag Berlin · Zbl 0668.54018 [30] Smyth, M. B., Totally bounded spaces and compact ordered spaces as domains of computation, (Reed, G. M.; Rosco, A. W.; Wachter, R. F., Topology and Category Theory in Computer Science (1991), Clarendon: Clarendon Oxford), 207-229 · Zbl 0733.54024 [31] Smyth, M. B., Completeness of quasi-uniform and syntopological spaces, J. London Math. Soc., 49, 385-400 (1994) · Zbl 0798.54036 [32] Sünderhauf, Ph., Quasi-uniform completeness in terms of Cauchy nets, Acta Math. Hungar., 69, 47-54 (1995) · Zbl 0845.54016 [33] Vitolo, P., A representation theorem for quasi-metric spaces, Topology Appl., 65, 101-104 (1995) · Zbl 0828.54024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.