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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 $\leq$ 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 [{\it G. T. Roberts}, Proc. Camb. Philos. Soc. 48, 533--546 (1952; Zbl 0047.10503)]. Moreover, the authors give some results concerning the definitions.

40J05Summability in abstract structures
40A35Ideal and statistical convergence
46A40Ordered topological linear spaces, vector lattices
Full Text: DOI
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