Convexity and w * -compactness in Banach spaces

“…They strengthen the original theorems and lead to new problems and applications in topology and analysis. See, for instance, [1,4,[8][9][10][11][12]. This paper goes along the same line, and it is organized as follows.…”

Section: Introductionmentioning

confidence: 93%

Distances to spaces of Baire one functions

2008

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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

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“…The topic of our theorem is part of the general question to which extent and in which sense co w * (B) can be recovered by a boundary B (cf. [2,3,4,5,6,9]). When K is a w * -compact subset of a bidual Banach space X * * and B = K, the distance dist(K, X) is, in general, different from the distance dist(co w * (K), X).…”

Section: ) (Which Equals 0 If X Is Reflexive and 1 If X Is Not Reflexmentioning

confidence: 99%