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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A lattice renorming theorem and applications to vector-valued processes

Authors: William J. Davis, Nassif Ghoussoub and Joram Lindenstrauss
Journal: Trans. Amer. Math. Soc. 263 (1981), 531-540
MSC: Primary 46B30; Secondary 60G99
MathSciNet review: 594424
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Abstract: A norm, $ \vert\vert\;\vert\vert$, on a Banach space $ E$ is said to be locally uniformly convex if $ \left\Vert {{x_n}} \right\Vert \to \left\Vert x \right\Vert$ and $ \left\Vert {{x_n} + x} \right\Vert \to 2\left\Vert x \right\Vert$ 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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Keywords: Banach lattice, local uniform convexity, renorming, vector-valued processes, martingales, submartingales, ergodic theorem
Article copyright: © Copyright 1981 American Mathematical Society