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Statistical convergence and statistical continuity on locally solid Riesz spaces. (English) Zbl 1244.40002

Let L be a real vector space and be a partial order on this space. The authors give the definition of an ordered vector space as follows:

(i) if x,yL and yx, then y+zx+z for each zL,

(ii) if x,yL and yx, then λyλx for each λ0·

In addition, if L is a lattice with respect to the partial ordering, then L is said to be a Riesz space (or a vector lattice).

Then they introduce the concepts of statistical topological convergence of a sequence, statistical τ-boundedness, statistical τ-Cauchy property, and statistical continuity in a locally solid Riesz space, which was introduced in [G. T. Roberts, Proc. Camb. Philos. Soc. 48, 533–546 (1952; Zbl 0047.10503)]. Moreover, the authors give some results concerning the definitions.

MSC:
40J05Summability in abstract structures
40A35Ideal and statistical convergence
46A40Ordered topological linear spaces, vector lattices
References:
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[9]Maio, G. D.; Kočinac, Lj.D.R.: Statistical convergence in topology, Topology appl. 156, 28-45 (2008) · Zbl 1155.54004 · doi:10.1016/j.topol.2008.01.015
[10]Mamedov, M. A.; Pehlivan, S.: Statistical cluster points and turnpike theorem in nonconvex problems, J. math. Anal. appl. 256, 686-693 (2001) · Zbl 1161.91452 · doi:10.1006/jmaa.2000.7061
[11]Riesz, F.: Sur la décomposition des opérations fonctionelles linéaires, , 143-148 (1930) · Zbl 56.0356.01
[12]Roberts, G. T.: Topologies in vector lattices, Math. proc. Cambridge philos. Soc. 48, 533-546 (1952) · Zbl 0047.10503
[13]Steinhaus, H.: Sur la convergence ordinarie et la convergence asymptotique, Colloq. math. 2, 73-74 (1951)
[14]Zaanen, A. C.: Introduction to operator theory in Riesz spaces, (1997)