A lattice renorming theorem and applications to vector-valued processes

Abstract

A norm, | | | | ||\;|| , on a Banach space E E is said to be locally uniformly convex if ‖ x n ‖ → ‖ x ‖ \left \| {{x_n}} \right \| \to \left \| x \right \| and ‖ x n + x ‖ → 2 ‖ x ‖ \left \| {{x_n} + x} \right \| \to 2\left \| x \right \| implies that x n → x {x_n} \to x in norm. It is shown that a Banach lattice has an (order) equivalent locally uniformly convex norm if and only if the lattice is order continuous. This result is used to reduce convergence theorems for (lattice-valued) positive martingales and submartingales to the scalar case.