Ideals of subseries convergence and F-spaces

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• 作者：Lech Drewnowski ; Iwo Labuda
• 关键词：F ; spaces ; Subseries convergence ; Unconditional convergence ; Ideal of sets ; \({F_{\sigma}}\) ; set ; \({F_{{\sigma}\delta}}\) ; set
• 刊名：Archiv der Mathematik
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：108
• 期：1
• 页码：55-64
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：1420-8938
• 卷排序：108

文摘

Let X be an F-space and \({\boldsymbol x=(x_n)}\) be a sequence of vectors in X. Ideals \({\mathcal{C}(\boldsymbol x)}\) of subseries convergence are considered. In particular, we show that a characterization of the class of Banach spaces not containing c₀ obtained by using the ideals \({\mathcal{C}(\boldsymbol x)}\) breaks down in every Fréchet space not isomorphic to a Banach space. On the other hand, the result can be extended to some F-spaces via the definition of a new class of F-spaces satisfying a stronger version of the condition (O) of Orlicz. A theorem discriminating between the finite and infinite dimensional case is obtained about the family \({\mathcal{C}(X)}\) of all ideals associated with the F-space X.