Coordinate systems and maximal order complexity in Banach lattices
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.