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

Authors:

• C. Angosto,
• Jerzy Kąkol,
• M. López-Pellicer

Abstract

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 U_n, and let H ⊂ E be a bounded set. Let ck(H):=sup{d(clust_E"(φ),E):φεH^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):hεH} the worst distance of H the weak *-closure in the bidual of H to E, where d means the natural metric of E". Let γ_n(H):=sup{|lim_plim_mu_p(h_m)-lim_mlim_pu_p(h_m)|:(u_p)⊂U_n⁰,(h_m)⊂H}, provided the involved limits exist. We extend a recent result of Angosto-Cascales to Fréchet spaces by showing that: If x^** εH, there is a sequence (x_p)_p in H such that d_n(x^**,y^**)≤γ_n(H) for each σ(E",E')-cluster point y^** of (x_p)_p and n ε 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)≤ d̂(H,C(X,Z))≤17ck(H), where H⊂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 d̂(H,C(X,Z)):=sup{f,C(X,Z),fεH}. This yields a quantitative version of Orihuela's angelic theorem. If X is strongly web-compact then ck(H)≤d̂(H,C(X,Z))≤5ck(H); this happens if X=(E',σ(E',E)) for EεG (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 d̂(H,C(X,Z))=ck(H) for each bounded H⊂R^X. © 2013 Elsevier Ltd.