On Normed Lattices and Their Banach Completions

On Normed Lattices and Their Banach Completions It is known that the Banach completion Y = bX of a normed lattice X need not preserve the properties to be Dedekind complete or σ-Dedekind complete. In this paper it is proved that the countable interpolation property and the property to be sequentially order complete are preserved under the Banach completion. To prove this results we found some sufficient conditions (which are close to necessary ones) on X which secure for Y to have the countable interpolation property and (respectively) to be sequentially order complete. These conditions are obtained with the help of the newly developed techniques based on representations of normed lattices. It is well known that order continuity, and σ-order continuity of a norm are preserved under the Banach completion. Here necessary and sufficient conditions on X to secure these properties in Y are discussed. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

On Normed Lattices and Their Banach Completions

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Kluwer Academic Publishers
Copyright © 2005 by Springer
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
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