On Quasibases and Bases of Symmetric Spaces Consisting of Nonnegative Functions

Based on the study of the geometric properties of unconditional quasibasic sequences, we show that in an arbitrary symmetric space there exists no unconditional quasibasis consisting of nonnegative functions. Moreover, we demonstrate that in an arbitrary Banach function lattice $X$ of type $p>1$ one can introduce an equivalent norm such that there exists no monotone (with respect to the new norm) basis in $X$ that consists of nonnegative functions.