On unconditional bases in certain Banach function spaces

摘要

Let X be a Banach space, L ∞([0,1])⊂X⊂L 1([0,1]), with an unconditional basis. By the well-known stability property in X, there exists a unconditional basis {f n} m=1 ∞ , where f n∈ in C([0,1]), n∈N. In this paper, we introduce the notion that X *has the singularity property of X *at a point t 0∈[0,1]. It is proved that if X *has the singularity property at a point t 0∈ [0,1], then there exists no orthonormal, fundamental system in C([0,1]) which forms an unconditional basis in X.