A quantitative approach to weak compactness in Fréchet spaces and spaces

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 c k ( H ) : = sup { d ( c l u s t E ′ ′ ( φ ) , E ) : φ ∈ 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 ∈ 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 γ n ( H ) : = sup { | lim p lim m u p ( h m ) − lim m lim p u p ( h m ) | : ( u p ) ⊂ U n 0 , ( h m ) ⊂ 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 ( 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 ∈ N . Moreover, k ( H ) = 0 iff c k ( H ) = 0 . This provides a quantitative version of the weak angelicity in a Fréchet space. Also we show that c k ( H ) ≤ d ˆ ( H ¯ , C ( X , Z ) ) ≤ 17 c k ( 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 c k ( 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 ˆ ( 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 c k ( H ) ≤ d ˆ ( H ¯ , C ( X , Z ) ) ≤ 5 c k ( H ) ; this happens if X = ( E ′ , σ ( 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 ˆ ( H ¯ , C ( X , Z ) ) = c k ( H ) for each bounded H ⊂ R X .