{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/22064"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/22064","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"On some spaces related to weak L(p) and their duals","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 $L\\sp{p})\\sp\\prime$.\"","abstract_html":""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 $L\\sp{p})\\sp\\prime$."","abstract_has_math":true,"creators":["Chung, Si Kit"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Lotz, Heinrich P."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T13:27:49Z","date_published":"2011-05-07T13:27:49Z","updated_at":"2026-07-22T22:25:19Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1993 Chung, Si Kit"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9411594","(UMI)AAI9411594"],"render_values":[{"text":"AAI9411594","href":null,"code":true},{"text":"(UMI)AAI9411594","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/22064","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Lotz, Heinrich P."]},{"key":"dc:creator","label":"Author","values":["Chung, Si Kit"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T13:27:49Z","10000-01-01","1993"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1993 Chung, Si Kit"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9411594","(UMI)AAI9411594","http://hdl.handle.net/2142/22064"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["\"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"]},{"key":"dc:title","label":"Title","values":["On some spaces related to weak L(p) and their duals"]}]}],"canonical_facts":{"dc:contributor":["Lotz, Heinrich P."],"dc:creator":["Chung, Si Kit"],"dc:date":["2011-05-07T13:27:49Z","10000-01-01","1993"],"dc:description":["\"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"],"dc:identifier":["AAI9411594","(UMI)AAI9411594","http://hdl.handle.net/2142/22064"],"dc:language":["eng"],"dc:rights":["Copyright 1993 Chung, Si Kit"],"dc:subject":["Mathematics"],"dc:title":["On some spaces related to weak L(p) and their duals"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:19Z"}