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Note on Extreme Points in Marcinkiewicz Function Spaces (Report)

By Anna Kaminska & Anca M. Parrish

Release : 2010-01-01

Genre : Mathematics, Books, Science & Nature

Kind : ebook

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1. INTRODUCTION AND PRELIMINARIES In [9, 10], Ryff considered extreme points of the convex set [OMEGA](w) of functions on [0,1] that is an orbit of a given function w. An orbit of a decreasing weight function w is in fact a unit ball of the Marcinkiewicz space [M.sub.W] corresponding to the weight w. Thus the Ryff's description can be applied directly to the characterization of extreme points of the unit ball of the Marcinkiewicz function space [M.sub.W] on the interval [0,1]. Further in [3], the analogous description has been given in the spaces of functions on the interval (0, [infinity]). Here we consider the Marcinkiewicz spaces [M.sub.W] over (0, [infinity]). We first show that the unit ball in the subspace [M.sup.0.sub.W] of all ordered continuous elements of [M.sub.W] has no extreme points. Moreover we provide a detailed proof, different than that given in [3, 9, 10], of the fact that f is an extreme point of the unit ball in [M.sub.W] if and only if [f.sup.*] = w.