An Extension of Kakutani's Theorem on Abstract Lp-spaces to the Case of any Positive p

An Extension of Kakutani's Theorem on Abstract Lp-spaces to the Case of any Positive p L -spaces to the Case of any Positive p S. TEERENSTRA Mathematical Institute, University of Nijmegen, Toernooiveld 1, 6525 ED Nijmegen, The Netherlands E-mail: 1. Introduction and Summary For p 2T1;1/, Kakutani’s theorem on abstract L -spaces (see [1, page 71]) gives a characterization of L -spaces within the class of Banach lattices. In the talk we presented an extension of this theorem that includes the case p 2 .0; 1/. The extension characterizes L -spaces for p 2 .0;1/, within the class of locally solid Riesz spaces (see [1, page 33]) and has the advantage of not distinguishing between the case p2T1;1/ (when L is a Banach lattice) and the case p 2 .0; 1/ (whenkk is no norm). Even for p 2T1;1/, the extension turns out to be a generalization, since the subadditivity ofkk is no longer required. 2. Motivation We first recall some definitions and properties relating to L -spaces. Let .S; A;/ be a measure space and p 2 .0;1/. Let M be the (Riesz) space of all measurable functions f V S ! R with identification of functions that are -almost everywhere equal and let L ./ be the p 1=p (Riesz) subspace Positivity Springer Journals

An Extension of Kakutani's Theorem on Abstract Lp-spaces to the Case of any Positive p

Positivity , Volume 4 (3) – Oct 16, 2004

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Kluwer Academic Publishers
Copyright © 2000 by Kluwer Academic Publishers
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
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