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

Let EE be a Fréchet space, i.e. a metrizable and complete locally convex space (lcs), E′′E′′ its strong second dual with a defining sequence of seminorms ‖⋅‖n‖⋅‖n induced by a decreasing basis of absolutely convex neighbourhoods of zero UnUn, and let H⊂EH⊂E be a bounded set. Let ck(H):=sup{d(clustE′′(φ),E):φ∈HN}ck(H):=sup{d(clustE′′(φ),E):φ∈HN} be the “worst” distance of the set of weak  ∗∗-cluster points in E″E″ of sequences in HH to EE, and k(H):=sup{d(h,E):h∈H¯} the worst distance of H¯ the weak  ∗∗-closure in the bidual of HH to EE, where dd means the natural metric of E′′E′′. Let γn(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(xp)p in HH such that dn(x∗∗,y∗∗)≤γn(H)dn(x∗∗,y∗∗)≤γn(H) for each σ(E′′,E′)σ(E′′,E′)-cluster point y∗∗y∗∗ of (xp)p(xp)p and n∈Nn∈N. Moreover, k(H)=0k(H)=0 iff ck(H)=0ck(H)=0. This provides a quantitative version of the weak angelicity in a Fréchet space. Also we show that ck(H)≤dˆ(H¯,C(X,Z))≤17ck(H), where H⊂ZXH⊂ZX is relatively compact and C(X,Z)C(X,Z) is the space of ZZ-valued continuous functions for a web-compact space XX and a separable metric space ZZ, being now ck(H)ck(H) the “worst” distance of the set of cluster points in ZXZX of sequences in HH to C(X,Z)C(X,Z), respect to the standard supremum metric dd, and dˆ(H¯,C(X,Z)):=sup{f,C(X,Z),f∈H¯}. This yields a quantitative version of Orihuela’s angelic theorem. If XX is strongly web-compact then ck(H)≤dˆ(H¯,C(X,Z))≤5ck(H); this happens if X=(E′,σ(E′,E))X=(E′,σ(E′,E)) for E∈GE∈G (for instance, if EE is a (DF)-space or an (LF)-space). In the particular case that EE is a separable metrizable locally convex space then dˆ(H¯,C(X,Z))=ck(H) for each bounded H⊂RXH⊂RX.