Unbounded norm convergence in Banach lattices
A net ( x α ) in a vector lattice X is unbounded order convergent to x ∈ X if | x α - x | ∧ u converges to 0 in order for all u ∈ X + . This convergence has been investigated and applied in several recent papers by Gao et al. It may be viewed as a generalization of almost everywhere convergence to g...
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| Veröffentlicht in: | Positivity : an international journal devoted to the theory and applications of positivity in analysis 2017-09, Vol.21 (3), p.963-974 |
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| container_title | Positivity : an international journal devoted to the theory and applications of positivity in analysis |
| container_volume | 21 |
| creator | Deng, Y. O’Brien, M. Troitsky, V. G. |
| description | A net
(
x
α
)
in a vector lattice
X
is unbounded order convergent to
x
∈
X
if
|
x
α
-
x
|
∧
u
converges to 0 in order for all
u
∈
X
+
. This convergence has been investigated and applied in several recent papers by Gao et al. It may be viewed as a generalization of almost everywhere convergence to general vector lattices. In this paper, we study a variation of this convergence for Banach lattices. A net
(
x
α
)
in a Banach lattice
X
is unbounded norm convergent to
x
if
for all
u
∈
X
+
. We show that this convergence may be viewed as a generalization of convergence in measure. We also investigate its relationship with other convergences. |
| doi_str_mv | 10.1007/s11117-016-0446-9 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_1931134098</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>1931134098</sourcerecordid><originalsourceid>FETCH-LOGICAL-c316t-b974932e7ca254860457019d460b1fe99bd6c27bfd5d88eeea908125b80ab253</originalsourceid><addsrcrecordid>eNp1kD1PwzAQhi0EEqXwA9giMRvuHDu2R6j4kiqxlNmyHaekap1ip0j8e1yFgYVb7ob3eU96CLlGuEUAeZexjKSADQXOG6pPyAyFZFQzhaflrpWgyDQ7Jxc5bwAKxWFG6Ht0wyG2oa3ikHaVH-JXSOsQfaj6WD3YaP1HtbXj2PuQL8lZZ7c5XP3uOVk9Pa4WL3T59vy6uF9SX2MzUqcl1zUL0lsmuGqACwmoW96Awy5o7drGM-m6VrRKhRCsBoVMOAXWMVHPyc1Uu0_D5yHk0WyGQ4rlo0FdI9YctCopnFI-DTmn0Jl96nc2fRsEc5RiJimmSDFHKUYXhk1MLtm4DulP87_QD6gLYpE</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>1931134098</pqid></control><display><type>article</type><title>Unbounded norm convergence in Banach lattices</title><source>Business Source Complete</source><source>SpringerLink Journals - AutoHoldings</source><creator>Deng, Y. ; O’Brien, M. ; Troitsky, V. G.</creator><creatorcontrib>Deng, Y. ; O’Brien, M. ; Troitsky, V. G.</creatorcontrib><description>A net
(
x
α
)
in a vector lattice
X
is unbounded order convergent to
x
∈
X
if
|
x
α
-
x
|
∧
u
converges to 0 in order for all
u
∈
X
+
. This convergence has been investigated and applied in several recent papers by Gao et al. It may be viewed as a generalization of almost everywhere convergence to general vector lattices. In this paper, we study a variation of this convergence for Banach lattices. A net
(
x
α
)
in a Banach lattice
X
is unbounded norm convergent to
x
if
for all
u
∈
X
+
. We show that this convergence may be viewed as a generalization of convergence in measure. We also investigate its relationship with other convergences.</description><identifier>ISSN: 1385-1292</identifier><identifier>EISSN: 1572-9281</identifier><identifier>DOI: 10.1007/s11117-016-0446-9</identifier><language>eng</language><publisher>Cham: Springer International Publishing</publisher><subject>Calculus of Variations and Optimal Control; Optimization ; Convergence ; Econometrics ; Fourier Analysis ; Lattices (mathematics) ; Mathematics ; Mathematics and Statistics ; Operator Theory ; Potential Theory ; Studies</subject><ispartof>Positivity : an international journal devoted to the theory and applications of positivity in analysis, 2017-09, Vol.21 (3), p.963-974</ispartof><rights>Springer International Publishing 2016</rights><rights>Positivity is a copyright of Springer, 2017.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c316t-b974932e7ca254860457019d460b1fe99bd6c27bfd5d88eeea908125b80ab253</citedby><cites>FETCH-LOGICAL-c316t-b974932e7ca254860457019d460b1fe99bd6c27bfd5d88eeea908125b80ab253</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s11117-016-0446-9$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s11117-016-0446-9$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Deng, Y.</creatorcontrib><creatorcontrib>O’Brien, M.</creatorcontrib><creatorcontrib>Troitsky, V. G.</creatorcontrib><title>Unbounded norm convergence in Banach lattices</title><title>Positivity : an international journal devoted to the theory and applications of positivity in analysis</title><addtitle>Positivity</addtitle><description>A net
(
x
α
)
in a vector lattice
X
is unbounded order convergent to
x
∈
X
if
|
x
α
-
x
|
∧
u
converges to 0 in order for all
u
∈
X
+
. This convergence has been investigated and applied in several recent papers by Gao et al. It may be viewed as a generalization of almost everywhere convergence to general vector lattices. In this paper, we study a variation of this convergence for Banach lattices. A net
(
x
α
)
in a Banach lattice
X
is unbounded norm convergent to
x
if
for all
u
∈
X
+
. We show that this convergence may be viewed as a generalization of convergence in measure. We also investigate its relationship with other convergences.</description><subject>Calculus of Variations and Optimal Control; Optimization</subject><subject>Convergence</subject><subject>Econometrics</subject><subject>Fourier Analysis</subject><subject>Lattices (mathematics)</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Operator Theory</subject><subject>Potential Theory</subject><subject>Studies</subject><issn>1385-1292</issn><issn>1572-9281</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2017</creationdate><recordtype>article</recordtype><sourceid>8G5</sourceid><sourceid>ABUWG</sourceid><sourceid>AFKRA</sourceid><sourceid>AZQEC</sourceid><sourceid>BENPR</sourceid><sourceid>CCPQU</sourceid><sourceid>DWQXO</sourceid><sourceid>GNUQQ</sourceid><sourceid>GUQSH</sourceid><sourceid>M2O</sourceid><recordid>eNp1kD1PwzAQhi0EEqXwA9giMRvuHDu2R6j4kiqxlNmyHaekap1ip0j8e1yFgYVb7ob3eU96CLlGuEUAeZexjKSADQXOG6pPyAyFZFQzhaflrpWgyDQ7Jxc5bwAKxWFG6Ht0wyG2oa3ikHaVH-JXSOsQfaj6WD3YaP1HtbXj2PuQL8lZZ7c5XP3uOVk9Pa4WL3T59vy6uF9SX2MzUqcl1zUL0lsmuGqACwmoW96Awy5o7drGM-m6VrRKhRCsBoVMOAXWMVHPyc1Uu0_D5yHk0WyGQ4rlo0FdI9YctCopnFI-DTmn0Jl96nc2fRsEc5RiJimmSDFHKUYXhk1MLtm4DulP87_QD6gLYpE</recordid><startdate>20170901</startdate><enddate>20170901</enddate><creator>Deng, Y.</creator><creator>O’Brien, M.</creator><creator>Troitsky, V. 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G.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c316t-b974932e7ca254860457019d460b1fe99bd6c27bfd5d88eeea908125b80ab253</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2017</creationdate><topic>Calculus of Variations and Optimal Control; Optimization</topic><topic>Convergence</topic><topic>Econometrics</topic><topic>Fourier Analysis</topic><topic>Lattices (mathematics)</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Operator Theory</topic><topic>Potential Theory</topic><topic>Studies</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Deng, Y.</creatorcontrib><creatorcontrib>O’Brien, M.</creatorcontrib><creatorcontrib>Troitsky, V. 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G.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Unbounded norm convergence in Banach lattices</atitle><jtitle>Positivity : an international journal devoted to the theory and applications of positivity in analysis</jtitle><stitle>Positivity</stitle><date>2017-09-01</date><risdate>2017</risdate><volume>21</volume><issue>3</issue><spage>963</spage><epage>974</epage><pages>963-974</pages><issn>1385-1292</issn><eissn>1572-9281</eissn><abstract>A net
(
x
α
)
in a vector lattice
X
is unbounded order convergent to
x
∈
X
if
|
x
α
-
x
|
∧
u
converges to 0 in order for all
u
∈
X
+
. This convergence has been investigated and applied in several recent papers by Gao et al. It may be viewed as a generalization of almost everywhere convergence to general vector lattices. In this paper, we study a variation of this convergence for Banach lattices. A net
(
x
α
)
in a Banach lattice
X
is unbounded norm convergent to
x
if
for all
u
∈
X
+
. We show that this convergence may be viewed as a generalization of convergence in measure. We also investigate its relationship with other convergences.</abstract><cop>Cham</cop><pub>Springer International Publishing</pub><doi>10.1007/s11117-016-0446-9</doi><tpages>12</tpages></addata></record> |
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| identifier | ISSN: 1385-1292 |
| ispartof | Positivity : an international journal devoted to the theory and applications of positivity in analysis, 2017-09, Vol.21 (3), p.963-974 |
| issn | 1385-1292 1572-9281 |
| language | eng |
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| source | Business Source Complete; SpringerLink Journals - AutoHoldings |
| subjects | Calculus of Variations and Optimal Control Optimization Convergence Econometrics Fourier Analysis Lattices (mathematics) Mathematics Mathematics and Statistics Operator Theory Potential Theory Studies |
| title | Unbounded norm convergence in Banach lattices |
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