Asymmetric norms given by symmetrisation and specialisation order.(English)Zbl 1394.54016

A function $$d:X\times X\to [0,\infty)$$ of a set $$X$$ is called a $$T_0$$-quasi-metric if the following conditions hold for all $$x,y,z\in X$$: $d(x,x)=0,$
$d(x,z)\leq d(x,y)+d(y,z),$
$d(x,y)=0=d(y,x) \text{ implies that }x=y.$
The authors in this paper continue the investigation between $$T_0$$-quasi-metric spaces and partially ordered metric spaces. In Section $$2$$, they show that it is possible to set up a Galois connection between these two classes of spaces (see Proposition $$2.2(c)$$). Furthermore, in Section $$3$$, the authors establish a relationship between nonexpansive maps between $$T_0$$-quasi-metric spaces and nonexpansive and increasing maps between partially ordered metric spaces (see Propositions $$3.1$$ and $$3.2$$).
Finally, in the last section (Section $$4$$), they derive a representation theorem for injective asymmetrically normed spaces. This answers a question that was left open in [J. Conradie et al., Topology Appl. 231, 92–112 (2017; Zbl 1387.46050)].

MSC:

 54E35 Metric spaces, metrizability 46M10 Projective and injective objects in functional analysis 46B40 Ordered normed spaces 54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces 06A15 Galois correspondences, closure operators (in relation to ordered sets)

Zbl 1387.46050
Full Text:

References:

 [1] Alegre, C.; Ferrando, I.; García-Raffi, L. M.; Sánchez Pérez, E. A., Compactness in asymmetric normed spaces, Topol. Appl., 155, 527-539, (2008) · Zbl 1142.46004 [2] Aliprantis, C. D.; Tourky, R., Cones and duality, Graduate Studies in Mathematics, vol. 84, (2007), American Mathematical Society Providence · Zbl 1127.46002 [3] Birkhoff, G., Lattice theory, Amer. Math. Soc. Colloq. Publ., vol. 25, (1967), American Mathematical Society Providence, RI · Zbl 0126.03801 [4] Conradie, J. J., Asymmetric norms, cones and partial orders, Topol. Appl., 193, 100-115, (2015) · Zbl 1344.46003 [5] Conradie, J. J.; Künzi, H-P. A.; Otafudu, O. O., The vector lattice structure on the isbell-convex hull of an asymmetrically normed real vector space, Topol. Appl., 231, 92-112, (2017) · Zbl 1387.46050 [6] Davie, B. A.; Priestley, H. A., Introduction to lattices and order, (2002), Cambridge University Press Cambridge · Zbl 1002.06001 [7] Gaba, Y. U.; Künzi, H-P. A., Splitting metrics by $$T_0$$-quasi-metrics, Topol. Appl., 193, 84-96, (2015) · Zbl 1345.54023 [8] Gaba, Y. U.; Künzi, H-P. A., Partially ordered metric spaces produced by $$T_0$$-quasi-metrics, Topol. Appl., 202, 366-383, (2016) · Zbl 1337.54022 [9] Kadison, R. V.; Ringrose, J. R., Fundamentals of the theory of operator algebras, vol. 1, (1983), Academic Press New York · Zbl 0518.46046 [10] Kemajou, E.; Künzi, H.-P. A.; Otafudu, O. O., The isbell-hull of a di-space, Topol. Appl., 159, 2463-2475, (2012) · Zbl 1245.54023 [11] Meyer-Nieberg, P., Banach lattices, (1991), Springer-Verlag Berlin · Zbl 0743.46015 [12] García-Raffi, L. M., Compactness and finite dimension in asymmetric normed linear spaces, Topol. Appl., 153, 844-854, (2005) · Zbl 1101.46017 [13] Riedl, J., Partially ordered locally convex vector spaces and extensions of positive continuous linear mappings, Math. Ann., 157, 95-124, (1964) · Zbl 0127.06403 [14] Schaefer, H. H., Topological vector spaces, (1971), Springer Berlin · Zbl 0217.16002
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.