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

Let $$L$$ be a real vector space and $$\leq$$ 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 $$y \leq x$$, then $$y+z\leq x+z$$ for each $$z\in L$$,
(ii) if $$x,y\in L$$ and $$y \leq x$$, then $$\lambda y\leq \lambda x$$ for each $$\lambda\geq 0.$$
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

Zbl 0047.10503
Full Text:

### References:

 [1] Aliprantis, C. D.; Burkinshaw, O., Locally Solid Riesz Spaces with Applications to Economics (2003), Amer. Math. Soc. · Zbl 1043.46003 [2] Buck, R. C., Generalized asymptotic density, Amer. J. Math., 75, 335-346 (1953) · Zbl 0050.05901 [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 [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), North-Holland: North-Holland Amsterdam · Zbl 0231.46014 [8] Maddox, I. J., Statistical convergence in a locally convex space, Math. Proc. Cambridge Philos. Soc., 104, 141-145 (1988) · Zbl 0674.40008 [9] Maio, G. D.; Kočinac, Lj. D.R., Statistical convergence in topology, Topology Appl., 156, 28-45 (2008) · Zbl 1155.54004 [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 [11] Riesz, F., Sur la décomposition des opérations fonctionelles linéaires, (Atti del Congr. Internaz. dei Mat., 3. Atti del Congr. Internaz. dei Mat., 3, Bologna, 1928 (1930), Zanichelli), 143-148 [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), Springer-Verlag · Zbl 0878.47022
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.