A lattice renorming theorem and applications to vector-valued processes

Davis, William J.; Ghoussoub, Nassif; Lindenstrauss, Joram

doi:10.1090/S0002-9947-1981-0594424-2

A lattice renorming theorem and applications to vector-valued processes
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by William J. Davis, Nassif Ghoussoub and Joram Lindenstrauss

Trans. Amer. Math. Soc. 263 (1981), 531-540

DOI: https://doi.org/10.1090/S0002-9947-1981-0594424-2

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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.

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