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Statements

Subject Item
n2:RIV%2F00216208%3A11320%2F12%3A10127317%21RIV13-AV0-11320___
rdf:type
n4:Vysledek skos:Concept
rdfs:seeAlso
http://dx.doi.org/10.1017/S0013091510000842
dcterms:description
We introduce two measures of weak non-compactness Ja_E and Ja that quantify, via distances, the idea of boundary that lies behind James's Compactness Theorem. These measures tell us, for a bounded subset C of a Banach space E and for given x* is an element of E*, how far from E or C one needs to go to find x** in w*-cl(C) with x**(x*) = sup x*(C). A quantitative version of James's Compactness Theorem is proved using Ja_E and Ja, and in particular it yields the following result. Let C be a closed convex bounded subset of a Banach space E and r > 0. If there is an element x_0** in w*-cl(C) whose distance to C is greater than r, then there is x* is an element of E* such that each x** is an element of w*-cl(C) at which sup x*(C) is attained has distance to E greater than 1/2 r. We indeed establish that Ja_E and Ja are equivalent to other measures of weak non-compactness studied in the literature. We also collect particular cases and examples showing when the inequalities between the different measures of weak non-compactness can be equalities and when the inequalities are sharp. We introduce two measures of weak non-compactness Ja_E and Ja that quantify, via distances, the idea of boundary that lies behind James's Compactness Theorem. These measures tell us, for a bounded subset C of a Banach space E and for given x* is an element of E*, how far from E or C one needs to go to find x** in w*-cl(C) with x**(x*) = sup x*(C). A quantitative version of James's Compactness Theorem is proved using Ja_E and Ja, and in particular it yields the following result. Let C be a closed convex bounded subset of a Banach space E and r > 0. If there is an element x_0** in w*-cl(C) whose distance to C is greater than r, then there is x* is an element of E* such that each x** is an element of w*-cl(C) at which sup x*(C) is attained has distance to E greater than 1/2 r. We indeed establish that Ja_E and Ja are equivalent to other measures of weak non-compactness studied in the literature. We also collect particular cases and examples showing when the inequalities between the different measures of weak non-compactness can be equalities and when the inequalities are sharp.
dcterms:title
A QUANTITATIVE VERSION OF JAMES'S COMPACTNESS THEOREM A QUANTITATIVE VERSION OF JAMES'S COMPACTNESS THEOREM
skos:prefLabel
A QUANTITATIVE VERSION OF JAMES'S COMPACTNESS THEOREM A QUANTITATIVE VERSION OF JAMES'S COMPACTNESS THEOREM
skos:notation
RIV/00216208:11320/12:10127317!RIV13-AV0-11320___
n4:predkladatel
n5:orjk%3A11320
n3:aktivita
n13:P n13:Z
n3:aktivity
P(IAA100190901), Z(MSM0021620839)
n3:cisloPeriodika
2
n3:dodaniDat
n21:2013
n3:domaciTvurceVysledku
n14:8577048 n14:3116085
n3:druhVysledku
n10:J
n3:duvernostUdaju
n18:S
n3:entitaPredkladatele
n22:predkladatel
n3:idSjednocenehoVysledku
120541
n3:idVysledku
RIV/00216208:11320/12:10127317
n3:jazykVysledku
n9:eng
n3:klicovaSlova
James's Compactness Theorem; measure of weak non-compactness; Banach space
n3:klicoveSlovo
n12:Banach%20space n12:James%27s%20Compactness%20Theorem n12:measure%20of%20weak%20non-compactness
n3:kodStatuVydavatele
GB - Spojené království Velké Británie a Severního Irska
n3:kontrolniKodProRIV
[08320EB5BCE3]
n3:nazevZdroje
Proceedings of the Edinburgh Mathematical Society
n3:obor
n20:BA
n3:pocetDomacichTvurcuVysledku
2
n3:pocetTvurcuVysledku
3
n3:projekt
n7:IAA100190901
n3:rokUplatneniVysledku
n21:2012
n3:svazekPeriodika
55
n3:tvurceVysledku
Kalenda, Ondřej Spurný, Jiří Cascales, Bernardo
n3:wos
000303129100006
n3:zamer
n15:MSM0021620839
s:issn
0013-0915
s:numberOfPages
18
n16:doi
10.1017/S0013091510000842
n17:organizacniJednotka
11320