Subspaces of asplund Banach spaces with the point continuity property

Boundedly complete weak-Cauchy basic sequences in Banach spaces with the PCP

“…Finally, we observe that if a Banach space B is spanned by a boundedly complete (s)-basis with difference sequence (e j ), and Y denotes the closed linear span of the e * j 's in B * , then the canonical map of B into Y * has range of codimension one (Proposition 6). These considerations then immediately yield the main result of Bellenot [1] and Finet [5]: if a Banach space X has separable dual and the PCP, then every non-trivial weak-Cauchy sequence in X has a subsequence spanning an order-one quasi-reflexive space (Corollary 6 below).…”

“…Our main result is proved using arguments along the lines of those in S. Bellenot [1] and C. Finet [5], and uses (as do the above authors) the fundamental result of N. Ghoussoub and B. Maurey [6] that every separable Banach space with the PCP has a boundedly complete skipped-blocking decomposition. We prove Theorem 1 by first observing in Proposition 2 that an (s)-sequence is boundedly complete if and only if its difference sequence is skippedboundedly complete.…”

On certain classes of Baire-1 functions with applications to Banach space theory

“…We note that if X is a Polish Banach space (i.e., Ba(X) is Polish in the weak topology) then Edgar and Wheeler [14] have shown that X is hereditarily reflexive (see also [37] and [18]). Bellenot [5] and Finet [15] have independently extended this result by showing that whenever X is Polish, if x * * ∈ X * * \ X then x * * | Ba(X * ) strictly governs the class of quasi-reflexive spaces of order 1.…”

Skipped Blocking and Other Decompositions in Banach Spaces

“…Ghoussoub and Maurey [7] using results form Edgar and Wheeler [5] showed a converse, separable spaces with PCP have a boundedly complete SBD. Other examples are in [14], [6], [1] and [2]. Having a boundedly complete SBD implies the existence of decompositions with stronger properties.…”