Coordinate systems and maximal order complexity in Banach lattices

Abstract

We resolve three questions arising from the study of coordinate systems and order convergence in Banach lattices. Firstly, every order (respectively, $σ$-order) basis $(e_n)$ for a Banach lattice $X=[e_n]$ is automatically a Schauder basis. In fact, the associated partial sum projections satisfy a maximal inequality, and hence every such basis is uniform. Secondly, we show that there is a separable Banach lattice $X$ in which the set $$ X_{\downarrow0}=\{(x_n)\in X^{\mathbb N}:x_1\geqslant x_2\geqslant\ldots\geqslant 0=\inf_nx_n\} $$ is $\mathbfΠ^1_1$-complete. Finally, we construct a separable Banach lattice $Y$ for which the set $Y^{\uparrow\infty}$ of increasing positive sequences which are not order bounded is $\mathbfΠ^1_1$-complete. It follows that neither $X_{\downarrow0}$ nor $Y^{\uparrow\infty}$ can be described in any simpler way than by the nonexistence of positive lower and upper bounds, respectively.