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

Abstract

It is proved that every non-trivial weak-Cauchy sequence in a Banach space with the PCP (the Point of Continuity Property) has a boundedly complete basic subsequence. The following result, due independently to S. Bellenot and C. Finet, is then deduced as a corollary. 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.

Weak-star point of continuity property and Schauder bases

“…As an easy consequence we also deduce from the above characterization of w * -PCP that every seminormalized basic sequence in a Banach space with PCP has a boundedly complete basic subsequence. This last result was obtained in [8].…”

“…The following consequence, obtained in a different way in [8], shows how many separable and dual subspaces contains every Banach space with PCP.…”

“…As a consequence of the above characterization we get that a dual space satisfies the Radon-Nikodym property if, and only if, every seminormalized topologically weak-star null tree has a boundedly complete branch, which improves some results in [3] obtained for the separable case. Also, as a consequence of the above characterization, the following result obtained in [8] is deduced: every seminormalized basic sequence in a Banach space with the point of continuity property has a boundedly complete subsequence.…”

Banach spaces with many boundedly complete basic sequences failing PCP

“…As PCP is separably determined [1], that is, a Banach space satisfies PCP if every separable subspace has PCP, it is natural looking for a sequential characterization of PCP. In this sense, it has been proved in [12] that every semi-normalized basic sequence in a Banach space with PCP has a boundedly complete subsequence. The converse of the above result is false in general, but it is open for Banach spaces not containing 1 (see Remark 2 in [12]).…”

“…In this sense, it has been proved in [12] that every semi-normalized basic sequence in a Banach space with PCP has a boundedly complete subsequence. The converse of the above result is false in general, but it is open for Banach spaces not containing 1 (see Remark 2 in [12]). The goal of this note is to prove in Corollary 2.4 that there exists a family of Banach spaces failing PCP and not containing 1 such that every semi-normalized basic sequence has a boundedly complete subsequence.…”