TVS-锥度量空间中的统计收敛
Statistical Convergence in TVS-cone Metric Spaces
本文研究TVS-锥度量空间中的统计收敛以及TVS-锥度量空间的统计完备性.令(X,E,P,d)表示一个TVS-锥度量空间.利用定义在有序Hausdorff拓扑向量空间E上的Minkowski函数ρ,证明了在X上存在一个通常意义下的度量dρ,使得X中的序列(xn)在锥度量d意义下统计收敛到x ∈ X,当且仅当(xn)在度量dρ意义下统计收敛到x.基于此,我们证明了任意一个TVS-锥统计Cauchy序列是几乎处处TVS-锥Cauchy序列,还证明了任意一个TVS-锥统计收敛的序列是几乎处处TVS-锥收敛的.从而,TVS-锥度量空间(X,d)是d-完备的,当且仅当它是d-统计完备的.基于以上结论,通常度量空间中统计收敛的许多性质都可以平行地推广到锥度量空间中统计收敛的情形.
The aims of this paper are to investigate the statistical convergence in TVScone metric spaces and to discuss statistical completeness of TVS-cone metric spaces. Let (X, E, P, d) be a TVS-cone metric space. By applying Minkowski function ρ in the ordered Hausdorff topological vector space E, we show that there exists a metric dρ (in usual sense) on X such that a sequence (xn) in X is statistically convergent to x ∈ X with respect to d if and only if it is statistically convergent to x with respect to dρ. We then show that every TVS-cone statistically Cauchy sequence is an almost usual TVS-cone Cauchy sequence, and every TVS-cone statistically convergent sequence is an almost usual TVS-cone convergent sequence. As a result, a TVS-cone metric space (X, d) is d-complete if and only if it is d-statistically complete. Based on the results obtained above, many properties of statistical convergence in the metric space can be generalized in parallel to the statistical convergence in the cone metric space.
统计收敛 / TVS-锥度量空间 / TVS-锥统计Cauchy序列 / Minkowski函数 / 统计完备 {{custom_keyword}} /
statistical convergence / TVS-cone metric space / TVS-cone statistically Cauchy sequence / Minkowski function / statistically complete {{custom_keyword}} /
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