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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 $\le$ be a partial order on this space. The authors give the definition of an ordered vector space as follows:

(i) if $x,y\in L$ and $y\le x$, then $y+z\le x+z$ for each $z\in L$,

(ii) if $x,y\in L$ and $y\le x$, then $\lambda y\le \lambda x$ for each $\lambda \ge 0·$

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 $\tau$-boundedness, statistical $\tau$-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:
 40J05 Summability in abstract structures 40A35 Ideal and statistical convergence 46A40 Ordered topological linear spaces, vector lattices
##### References:
 [1] Aliprantis, C. D.; Burkinshaw, O.: Locally solid Riesz spaces with applications to economics, (2003) [2] Buck, R. C.: Generalized asymptotic density, Amer. J. Math. 75, 335-346 (1953) · Zbl 0050.05901 · doi:10.2307/2372456 [3] Fast, H.: Sur la convergence statistique, Colloq. math. 2, 241-244 (1951) · Zbl 0044.33605 [4] Fridy, J. A.: On statistical convergence, Analysis 5, 301-313 (1985) · Zbl 0588.40001 [5] Fridy, J. A.: Statistical limit points, Proc. amer. Math. soc. 118, 1187-1192 (1993) · Zbl 0776.40001 · doi:10.2307/2160076 [6] Kantorovich, L. V.: Lineare halbgeordnete raume, Rec. math. 2, 121-168 (1937) · Zbl 0016.40502 [7] Luxemburg, W. A. J.; Zaanen, A. C.: Riesz spaces I, (1971) [8] Maddox, I. J.: Statistical convergence in a locally convex space, Math. proc. Cambridge philos. Soc. 104, 141-145 (1988) · Zbl 0674.40008 · doi:10.1017/S0305004100065312 [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)