SUBSPACES CONTAINING BIORTHOGONAL FUNCTIONALS OF BASES OF DIFFERENT TYPES M.I.Ostrovskii

The paper is devoted to two particular cases of the following general problem. Let α and β be two types of bases in Banach spaces. Let a Banach space X has bases of both types and a subspace M ⊂ X∗ contains the sequence of biorthogonal functionals of some α-basis in X. Does M contain a sequence of biorthogonal functionals of some β-basis in X? The following particular cases are considered: (α, β)=(Schauder bases, unconditional bases), (α, β)=(Nonlinear operational bases, linear operational bases). The paper contains an investigation of some of the spaces constructed by S.Bellenot in “The J-sum of Banach spaces”, J. Funct. Anal. 48 (1982), 95–106. (These spaces are used in some examples.) We use the standard Banach space notation as can be found in [LT2], [PP], [S2]. 1. Definition 1. Let X be a Banach space with (unconditional) basis. A subspace M ⊂ X is called (unconditionally) basic if it contains all biorthogonal functionals of some (unconditional) basis of X . Basic subspaces have been studied in [DK], [O2]. Theorem 1. Let X be a non-reflexive Banach space with an unconditional basis. There exists a subspace of X which is basic but is not unconditionally basic. Proof. Let (xi) ∞ i=1 be an unconditional basis of X and x ∗ i (i ∈ N) be its biorthogonal functionals. Then either (xi) is boundedly complete or it is not. Suppose first that (xi) is boundedly complete. Then X can be equivalently renormed to become the dual of the space N = [xi ] ∞ i=1, in natural duality. (We use square brackets to denote the closure of linear span.) The space N is a non-reflexive Banach space and (xi ) is an unconditional shrinking basis of it. So by James’ theorems [LT2, p. 9, 22] this basis is not boundedly complete and the space N contains a sequence of blocks