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Title:  On some spaces related to weak L(p) and their duals 
Author(s):  Chung, Si Kit 
Doctoral Committee Chair(s):  Lotz, Heinrich P. 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
Abstract:  In Cwikel's paper "On the dual of Weak $L\sp{p}$", it is shown that (Weak $L\sp{p})\sp\prime = L(p\sp\prime, 1)\oplus S\sb0\oplus S\sb\infty.$ For nonatomic measure spaces, Cwikel obtains a representation of elements in $S\sb0$ and $S\sb\infty.$ However, we show that this representation is incorrect by proving that if E is a nonreflexive weakly sequentially complete Banach lattice, then the disjoint complement of E in $E\sp{\prime\prime}$ is nonreflexive. So, we would like to obtain more information on (Weak $L\sp{p})\sp\prime$. For simplicity, we consider the Lebesgue measure space on (0,1) so that $S\sb\infty=\{0\}.$ We introduce three lattice seminorms $\rho\sb0,\rho\sb1$ and $\rho\sb\omega$ on $L\sp\infty(0,1)$ so that ($L\sp\infty(0,1),\rho\sb\omega)$ can be identified as an ideal of a quotient of Weak $L\sp\rho(0,1).$ The dual of $(L\sp\infty(0,1),\rho\sb{i})$ where i = 0, 1, $\omega$ is studied using the result that if E is a normed vector lattice and ${\cal A}$ is a bounded subset of $E\sb+\sp\prime,$ then the unit ball of $(E,\rho\sb{\cal A})\sp\prime$ is the solid hull of the $\sigma(E\sp\prime, E)$closed convex hull generated by ${\cal A},$ where $\rho\sb{\cal A}$ is the lattice seminorm defined by $\rho\sb{\cal A}(x) = \sup\sb{x\sp\prime\in {\cal A}}\langle \vert x\vert, x\sp\prime\rangle.$ We prove that the maximal elements in the unit ball of $(L\sp\infty(0,1),\rho\sb0)\sp\prime$ are the nonincreasing means concentrated at 0 and that these elements are weak*limits of nets of nonincreasing, nonnegative functions with $L\sb1$norms equal to one and supports shrinking to 0. We introduce the idea of dual admissibility of an ordered pair $(\Vert\cdot\Vert\sb1,\Vert\cdot\Vert\sb0)$ of lattice norms defined on a vector lattice. Characterizations of and sufficient conditions for dual admissibility are obtained. From this, we show that the unit ball of ($L\sp\infty(0,1),\rho\sb{\omega})\sp\prime$ can be obtained by taking the weak*closure in $L\sp\infty(0,1)\sp\prime$ of a certain subset of $(L\sp\infty(0,1),\rho\sb0)\sp\prime.$ Then we show that every element in $(L\sp\infty(0,1),\rho\sb{\omega})\sp\prime$ has a unique norm preserving "extension" in $S\sb0$ and that $S\sb0$ can be "generated" by these norm preserving "extensions" together with a family of operators. Finally, we consider questions related to dual admissibility. Results on the equivalence of order continuous norm topologies on order intervals as well as that on the $\sigma(E\sp\prime,E)$density in $E\sbsp{+}{\prime}$ of the positive part of a sublattice of the dual of a normed vector lattice E are obtained. 
Issue Date:  1993 
Type:  Text 
Language:  English 
URI:  http://hdl.handle.net/2142/22064 
Rights Information:  Copyright 1993 Chung, Si Kit 
Date Available in IDEALS:  20110507 
Identifier in Online Catalog:  AAI9411594 
OCLC Identifier:  (UMI)AAI9411594 
This item appears in the following Collection(s)

Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois