Abstract
We investigate partially ordered normed vector spaces in which the norm convergence coincides with the order convergence. We consider spaces where the convergences coincide for arbitrary nets and spaces where the convergences coincide only for sequences. We give conditions which characterize such spaces and investigate their properties. In particular, we study the problem of their Dedekind completeness and σ-completeness.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literature cited
R. E. De Marr, “Order convergence in linear topological spaces,” Pacific J. Math.,14, No. 1, 17–20 (1964).
B. Z. Vulikh and I. F. Danilenko, “On a method of partial ordering of normed spaces,” Vestnik LGU, No. 19, 18–22 (1970).
M. A. Krasnosel'skii, Positive Solutions of Operator Equations, P. Noordhoff, Groningen (1964).
A. M. Rubinov, “Infinite dimensional production models,” Sibirsk. Matem. Zh.,10, No. 6, 1375–1386 (1969).
V. A. Solov'ev, “On the extension of a monotone norm from a normed lattice to its Dedekind completion,” Sibirsk. Matem. Zh.,7, No. 6, 1360–1369 (1966).
B. Z. Vulikh, Introduction to the Theory of Partially Ordered Spaces, Wolters-Noordhoff, Groningen (1967).
E. E. Gurevich and G. Ya. Rotkovich, “The comparison of the different definitions of (o)-convergence in lattices,” Uchen. Zap. Leningrad. Ped. Inst.,274, 52–58 (1965).
Author information
Authors and Affiliations
Additional information
Translated from Matematicheskie Zametki, Vol. 13, No. 2, pp. 259–268, February, 1973.
Rights and permissions
About this article
Cite this article
Vulikh, B.Z., Korsakova, O.S. On spaces in which the norm convergence coincides with the order convergence. Mathematical Notes of the Academy of Sciences of the USSR 13, 158–163 (1973). https://doi.org/10.1007/BF01094235
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01094235