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Chung, Si Kit 2011-05-07T13:27:49Z Lotz, Heinrich P. 2011-05-07T13:27:49Z 10000-01-01 1993 "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. Made available in DSpace on 2011-05-07T13:27:49Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9411594.pdf: 2814290 bytes, checksum: 14dc962156cd38fd30983644d8cfaeec (MD5) Previous issue date: 1993 Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:55:04Z Item is restricted indefinitely. Restriction data tranferred 2014-07-01T11:25:39-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission ETDs are only available to UIUC Users without author permission U of I Only AAI9411594 (UMI)AAI9411594 http://hdl.handle.net/2142/22064 eng Copyright 1993 Chung, Si Kit Mathematics On some spaces related to weak L(p) and their duals text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.