Let X be a real Banach space. A subset B of the dual unit sphere of X is said to be a boundary for X, if every element of X attains its norm on some functional in B. The well-known Boundary Problem originally posed by Godefroy asks whether a bounded subset of X which is compact in the topology of pointwise convergence on B is already weakly compact. This problem was recently solved by Pfitzner in the positive. In this note we collect some stronger versions of the solution to the Boundary Problem, most of which are restricted to special types of Banach spaces. We shall use the results and techniques of Pfitzner, Cascales et al., Moors and others.
We establish here some inequalities between distances of pointwise bounded subsets H of R X to the space of real-valued continuous functions C(X) that allow us to examine the quantitative difference between (pointwise) countable compactness and compactness of H relative to C(X). We prove, amongst other things, that if X is a countably K-determined space the worst distance of the pointwise closure H of H to C(X) is at most 5 times the worst distance of the sets of cluster points of sequences in H to C(X): here distance refers to the metric of uniform convergence in R X . We study the quantitative behavior of sequences in H approximating points in H . As a particular case we obtain the results known about angelicity for these C p (X) spaces obtained by Orihuela. We indeed prove our results for spaces C(X, Z) (hence for Banach-valued functions) and we give examples that show when our estimates are sharp.
We extend the result of B. Cascales at al. about expand-contract plasticity of the unit ball of strictly convex Banach space to those spaces whose unit ball is the union of all its finitedimensional polyhedral extreme subsets. We also extend the definition of expand-contract plasticity to uniform spaces and generalize the theorem on expand-contract plasticity of totally bounded metric spaces to this new setting.
Given a metric space X and a Banach space (E, · ) we use an index of σ -fragmentability for maps f ∈ E X to estimate the distance of f to the space B 1 (X, E) of Baire one functions from X into (E, · ). When X is Polish we use our estimations for these distances to give a quantitative version of the well known Rosenthal's result stating that in B 1 (X, R) the pointwise relatively countably compact sets are pointwise relatively compact. We also obtain a quantitative version of a Srivatsa's result that states that whenever X is metric any weakly continuous function f ∈ E X belongs to B 1 (X, E): our result here says that for an arbitrary f ∈ E X we have
Main result Let E be a Banach space and let H ⊂ E be a bounded subset of E. Then d(co(H), E) ≤ 2 d(H, E), closures are weak *-closures taken in the bidual E * * ; d(A, E) := sup{d(a, E) : a ∈ A} for A ⊂ E * * ; d(A, E) = 0 iff A ⊂ E. Hence the inequality implies Krein's theorem (if H is relatively weakly compact then co(H) is weakly compact.) B. Cascales Compactness+Distances The starting point.. . our goals The results References The starting point.. .
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