A non-linear method for constructing certain basic sequences in Banach spaces

Abstract

isomorphic to an infinite dimensional conjugate space. This is, of course, a first step toward the better known conjecture that every Banach space contains an isomorphic copy of c o or or an infinite dimensional reflexive subspace. In this paper, we exhibit a new technique for constructing infinite boundedly complete basic sequences and consequently separable conjugate subspaces. Aninteresting feature of this method is that it involves a non-linear approach, which is in sharp contrast with the linear nature of the problem we address. Our approach also combines techniques originating in the local theory of Banach spaces (Dvoretzky’s theorem and concentration phenomenon) with infinite dimensional concepts like dentability and transfinite slicing of sets. One aPllication of this method is that Banach spaces with the Analytic Radon-Nikbdym Property (ARNP) contain copies of infinite dimensional conjugate spaces. (Recall that a complex Banach space X is said to have the ARNP if every X-valued bounded analytic map on the open unit disc of the complex plane has radial limits almost surely). This class of spaces containsbesides those possessing the Radon-Nikodym property (RNP) (see [D-U])--ali Banach lattices not containing c o [B-D] as well as all preduals of Von Neuman algebras [H-P]. What is needed for the proof is the following geometric characterization of such spaces established in [G-L-M]: Every bounded subset of a Banach space with the ARNP has arbitrarily norm-small determined by Lipschitz and plurisubharmonic functions. In the classical RNP setting (where the are determined by continuous linear functionals) the analogous statement (i.e. The existence of infinite dimensional conjugate spaces) was established in [G-M1]. Also shown there is the case where the slices are determined by a finite number of linear functionals i.e.