About: A QUANTITATIVE VERSION OF JAMES'S COMPACTNESS THEOREM

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rdf:type	skos:Concept http://linked.opendata.cz/ontology/domain/vavai/Vysledek
rdfs:seeAlso	http://dx.doi.org/10.1017/S0013091510000842
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. (en)
Title	A QUANTITATIVE VERSION OF JAMES'S COMPACTNESS THEOREM A QUANTITATIVE VERSION OF JAMES'S COMPACTNESS THEOREM (en)
skos:prefLabel	A QUANTITATIVE VERSION OF JAMES'S COMPACTNESS THEOREM A QUANTITATIVE VERSION OF JAMES'S COMPACTNESS THEOREM (en)
skos:notation	RIV/00216208:11320/12:10127317!RIV13-AV0-11320___
http://linked.open...avai/riv/aktivita	P Z
http://linked.open...avai/riv/aktivity	P(IAA100190901), Z(MSM0021620839)
http://linked.open...iv/cisloPeriodika	2
http://linked.open...vai/riv/dodaniDat	2013
http://linked.open...aciTvurceVysledku	Spurný, Jiří Kalenda, Ondřej
http://linked.open.../riv/druhVysledku	J - Článek v odborném periodiku
http://linked.open...iv/duvernostUdaju	S - Úplné a pravdivé údaje nepodléhající ochraně podle zvláštních právních předpisů
http://linked.open...titaPredkladatele	Univerzita Karlova v Praze / Matematicko-fyzikální fakulta
http://linked.open...dnocenehoVysledku	120541
http://linked.open...ai/riv/idVysledku	RIV/00216208:11320/12:10127317
http://linked.open...riv/jazykVysledku	eng - angličtina
http://linked.open.../riv/klicovaSlova	James's Compactness Theorem; measure of weak non-compactness; Banach space (en)
http://linked.open.../riv/klicoveSlovo	James's Compactness Theorem measure of weak non-compactness Banach space
http://linked.open...odStatuVydavatele	GB - Spojené království Velké Británie a Severního Irska
http://linked.open...ontrolniKodProRIV	[08320EB5BCE3]
http://linked.open...i/riv/nazevZdroje	Proceedings of the Edinburgh Mathematical Society
http://linked.open...in/vavai/riv/obor	BA
http://linked.open...ichTvurcuVysledku	2 (xsd:int)
http://linked.open...cetTvurcuVysledku	3 (xsd:int)
http://linked.open...vavai/riv/projekt	Topological and geometric structures in Banach spaces
http://linked.open...UplatneniVysledku	2012
http://linked.open...v/svazekPeriodika	55
http://linked.open...iv/tvurceVysledku	Kalenda, Ondřej Spurný, Jiří Cascales, Bernardo
http://linked.open...ain/vavai/riv/wos	000303129100006
http://linked.open...n/vavai/riv/zamer	Metody moderní matematiky a jejich aplikace
issn	0013-0915
number of pages	18 (xsd:int)
http://bibframe.org/vocab/doi	10.1017/S0013091510000842
http://localhost/t...ganizacniJednotka	11320

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