A quantitative approach to weak compactness in Fr¨¦chet spaces and spaces

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• 作者：C. Angosto^a ; ^{carlos.angosto@upct.es} ; J. Ka?kol^b ; ^{kakol@amu.edu.pl} ; M. Ló ; pez-Pellicer^c ; ^{mlopezpe@mat.upv.es}
• 关键词：Angelicity ; Compact space ; Countably compact ; C(X)C(X) ; Fré ; chet space ; Web compact spaces
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2013
• 出版时间：1 July, 2013
• 年：2013
• 卷：403
• 期：1
• 页码：13-22
• 全文大小：432 K

文摘

Let be a Fr¨¦chet space, i.e. a metrizable and complete locally convex space (lcs), its strong second dual with a defining sequence of seminorms induced by a decreasing basis of absolutely convex neighbourhoods of zero , and let be a bounded set. Let be the ¡°worst¡± distance of the set of weak -cluster points in of sequences in to , and the worst distance of the weak -closure in the bidual of to , where means the natural metric of . Let , provided the involved limits exist. We extend a recent result of Angosto-Cascales to Fr¨¦chet spaces by showing that: If , there is a sequence in such that for each -cluster point of and . Moreover, iff . This provides a quantitative version of the weak angelicity in a Fr¨¦chet space. Also we show that , where is relatively compact and is the space of -valued continuous functions for a web-compact space and a separable metric space , being now the ¡°worst¡± distance of the set of cluster points in of sequences in to , respect to the standard supremum metric , and . This yields a quantitative version of Orihuela¡¯s angelic theorem. If is strongly web-compact then ; this happens if for (for instance, if is a (DF)-space or an (LF)-space). In the particular case that is a separable metrizable locally convex space then for each bounded .