On some spaces related to weak L(p) and their duals

Chung, Si Kit

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https://hdl.handle.net/2142/22064 Copy

Description

Title

On some spaces related to weak L(p) and their duals

Author(s)

Chung, Si Kit

Issue Date

1993

Doctoral Committee Chair(s)

Lotz, Heinrich P.

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

Mathematics

Language

eng

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 non-atomic 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 non-reflexive weakly sequentially complete Banach lattice, then the disjoint complement of E in $E\sp{\prime\prime}$ is non-reflexive. 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 semi-norms $\rho\sb0,\rho\sb1$ 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 semi-norm 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 non-increasing means concentrated at 0 and that these elements are weak*-limits of nets of non-increasing, non-negative 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.

Type of Resource

text

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Copyright 1993 Chung, Si Kit

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