The canonical partial metric and the uniform convexity on normed spaces

S. Oltra, Salvador Romaguera, E.A. Sánchez-Pérez

Abstract

In this paper we introduce the notion of canonical partial metric associated to a norm to study geometric properties of normed spaces. In particular, we characterize strict convexity and uniform convexity of normed spaces in terms of the canonical partial metric defined by its norm.

We prove that these geometric properties can be considered, in this sense, as topological properties that appear when we compare the natural metric topology of the space with the non translation invariant topology induced by the canonical partial metric in the normed space.


Keywords

Partial metric; Convexity; Normed spaces

Full Text:

PDF

References

B. Beauzamy, Introduction to Banach Spaces and their Geometry, North Holland Math. Studies, Amsterdam (1985).

Á. Császár, Fondements de la Topologie Générale, Budapest-Paris (1960).

P. Fletcher and W. F. Lindgren, Quasi-uniform Spaces, Marcel Dekker, New York (1982).

H. P. A. Künzi, Nonsymmetric topology, in: Proc. Szekszárd Conference, Bolyai Soc. Math. Studies 4 1993 Hungary (Budapest 1995), 303–338.

J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II, Springer, Berlin (1996).

S. G. Matthews, Partial metric topology, in: Proc. 8th Summer Conference on General Topology and Applications. Ann. New York Acad. Sci. 728 (1994), 183-197. http://dx.doi.org/10.1111/j.1749-6632.1994.tb44144.x

S. J. O’Neill, Partial metrics, valuations and domain theory, in: Proc. 11th Summer Conference on General Topology and Applications. Ann. New York Acad. Sci. 806 (1996), 304–315. http://dx.doi.org/10.1111/j.1749-6632.1996.tb49177.x

S. Oltra, S. Romaguera and E. A. Sánchez-Pérez, Bicompleting weightable quasi-metric spaces and partial metric spaces, Rend. Circ. Mat. Palermo. Serie II, T.LI (2002), 151–162.

S. Oltra and E. A. Sánchez-Pérez, Order properties and p-metrics on Köthe function spaces, Houston J. Math., to appear.

W. Rudin, Functional Analysis, McGraw-Hill, New York (1973).

P. Wojtaszczyk, Banach Spaces for Analysts, Cambridge University Press, Cambridge (1991).

Abstract Views

979
Metrics Loading ...

Metrics powered by PLOS ALM


 

Cited-By (articles included in Crossref)

This journal is a Crossref Cited-by Linking member. This list shows the references that citing the article automatically, if there are. For more information about the system please visit Crossref site

1. Fixed point results for mappings satisfying (ψ φ)-weakly contractive condition in ordered partial metric spaces
Hemant Nashine
Mathematica Slovaca  vol: 63  issue: 4  year: 2013  
doi: 10.2478/s12175-013-0141-2

2. Uniform structures in the beginning of the third millenium
Hans-Peter A. Künzi
Topology and its Applications  vol: 154  issue: 14  first page: 2745  year: 2007  
doi: 10.1016/j.topol.2007.05.008



Esta revista se publica bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.

Universitat Politècnica de València

e-ISSN: 1989-4147   https://doi.org/10.4995/agt