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

作者：Angosto C; Kakol J; Lopez Pellicer M^*

来源：Journal of Mathematical Analysis and Applications, 2013, 403(1): 13-22.

DOI：10.1016/j.jmaa.2013.01.055

• Angelicity
• Compact space
• Countably compact
• C(X)
• Frechet space
• Web compact spaces

Let E be a Frechet space, i.e. a metrizable and complete locally convex space (lcs), E %26apos;%26apos; its strong second dual with a defining sequence of seminorms parallel to center dot parallel to(n) induced by a decreasing basis of absolutely convex neighbourhoods of zero U-n, and let H subset of E be a bounded set. Let ck(H) := sup{d(cluste(E %26apos;%26apos;) (phi), E) : phi is an element of H-N} be the %26quot;worst%26quot; distance of the set of weak *-cluster points in E %26apos;%26apos; of sequences in H to E, and k(H) := sup{d(h, E) : h is an element of (H) over bar} the worst distance of (H) over bar the weak *-closure in the bidual of H to E, where d means the natural metric of E %26apos;%26apos;. Let gamma(n)(H) := sup {vertical bar lim(p) lim(m) u(p) (h(m)) - lim(m) lim(p) u(p) (h(m))vertical bar : (u(p)) subset of U-n(0), (h(m)) subset of H}, provided the involved limits exist. We extend a recent result of Angosto-Cascales to Frechet spaces by showing that: If x** is an element of (H) over bar, there is a sequence (x(p))(p) in H such that d(n)(x**, y**) %26lt;= gamma(n)(H) for each sigma (E %26apos;%26apos;, E%26apos;)-cluster point y** of (x(p))(p) and n is an element of N. Moreover, k(H) = 0 iff ck(H) = 0. This provides a quantitative version of the weak angelicity in a Frechet space. Also we show that ck(H) %26lt;= (d) over cap((H) over bar, C(X, Z)) %26lt;= 17ck(H), where H subset of Z(X) 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 %26quot;worst%26quot; distance of the set of cluster points in Z(X) of sequences in H to C(X, Z), respect to the standard supremum metric d, and (d) over cap((H) over bar, C(X, Z)) := sup{f, C(X, Z), f is an element of (H) over bar}. This yields a quantitative version of Orihuela%26apos;s angelic theorem. If X is strongly web-compact then ck(H) %26lt;= (d) over cap((H) over bar, C(X, Z)) %26lt;= 5ck(H); this happens if X = (E%26apos;, sigma(E%26apos;, E)) for E is an element of (sic) (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) over cap((H) over bar, C(X, Z)) = ck(H) for each bounded H subset of R-X.

• 出版日期2013-7-1