A quantitative approach to weak compactness in Fréchet spaces and spaces C(X)

Journal ar
Journal of Mathematical Analysis and Applications
  • Volumen: 403
  • Número: 1
  • Fecha: 01 July 2013
  • Páginas: 13-22
  • ISSN: 0022247X 10960813
  • Source Type: Journal
  • DOI: 10.1016/j.jmaa.2013.01.055
  • Document Type: Article
Let E be a Fréchet space, i.e. a metrizable and complete locally convex space (lcs), E" its strong second dual with a defining sequence of seminorms ||.||n induced by a decreasing basis of absolutely convex neighbourhoods of zero Un, and let H ¿ E be a bounded set. Let ck(H):=sup{d(clustE"(¿),E):¿¿HN} be the "worst" distance of the set of weak *-cluster points in E" of sequences in H to E, and k(H):=sup{d(h,E):h¿H} the worst distance of H the weak *-closure in the bidual of H to E, where d means the natural metric of E". Let ¿n(H):=sup{|limplimmup(hm)-limmlimpup(hm)|:(up)¿Un0,(hm)¿H}, provided the involved limits exist. We extend a recent result of Angosto-Cascales to Fréchet spaces by showing that: If x** ¿H, there is a sequence (xp)p in H such that dn(x**,y**)¿¿n(H) for each ¿(E",E')-cluster point y** of (xp)p and n ¿ N. Moreover, k(H)=0 iff ck(H)=0. This provides a quantitative version of the weak angelicity in a Fréchet space. Also we show that ck(H)¿ d¿(H,C(X,Z))¿17ck(H), where H¿ZX is relatively compact and C(X,Z) is the space of Z-valued continuous functions for a web-compact space X and a separable metric space Z, being now ck(H) the "worst" distance of the set of cluster points in ZX of sequences in H to C(X,Z), respect to the standard supremum metric d, and d¿(H,C(X,Z)):=sup{f,C(X,Z),f¿H}. This yields a quantitative version of Orihuela's angelic theorem. If X is strongly web-compact then ck(H)¿d¿(H,C(X,Z))¿5ck(H); this happens if X=(E',¿(E',E)) for E¿G (for instance, if E is a (DF)-space or an (LF)-space). In the particular case that E is a separable metrizable locally convex space then d¿(H,C(X,Z))=ck(H) for each bounded H¿RX. © 2013 Elsevier Ltd.

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