A Lattice Renorming Theorem and Applications to Vector-Valued Processes

Abstract

A norm, $||\;||$, on a Banach space $E$ is said to be locally uniformly convex if $\left \| {{x_n}} \right \| \to \left \| x \right \|$ and $\left \| {{x_n} + x} \right \| \to 2\left \| x \right \|$ implies that ${x_n} \to x$ 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.