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

Abstract

[EN] Let E be a Frechet space, i.e. a metrizable and complete locally convex space (lcs), E '' 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 '') (phi), E) : phi is an element of H-N} 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 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 ''. 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**) <= gamma(n)(H) for each sigma (E '', E')-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) <= (d) over cap((H) over bar, C(X, Z)) <= 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 "worst" 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's angelic theorem. If X is strongly web-compact then ck(H) <= (d) over cap((H) over bar, C(X, Z)) <= 5ck(H); this happens if X = (E', sigma(E', 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

[ES] Se obtienen medidas que caracterizan cuantitativamente la compacidad débil en espacios de Fréchet. De estas medidas se deducen pruebas muy sencillas de resultados de compacidad en espacios de Fréchet, extendiendo resultados previos obtenidos recientemente en espacios de Banach. Además se obtienen medidas cuantitativas de compacidad en espacios C(X) con la topología puntual, estudiando la aproximación por sucesiones, así como diferentes propiedades del espacio de funciones continuas C(X) para clases importantes de espacios X.