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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 [G. T. Roberts, Proc. Camb. Philos. Soc. 48, 533–546 (1952; Zbl 0047.10503)]. Moreover, the authors give some results concerning the definitions.

40J05 Summability in abstract structures (should also be assigned at least one other classification number from Section 40-XX)
40A35 Ideal and statistical convergence
46A40 Ordered topological linear spaces, vector lattices
Full Text: DOI
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