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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)
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Title
| - A QUANTITATIVE VERSION OF JAMES'S COMPACTNESS THEOREM
- A QUANTITATIVE VERSION OF JAMES'S COMPACTNESS THEOREM (en)
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skos:prefLabel
| - A QUANTITATIVE VERSION OF JAMES'S COMPACTNESS THEOREM
- A QUANTITATIVE VERSION OF JAMES'S COMPACTNESS THEOREM (en)
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skos:notation
| - RIV/00216208:11320/12:10127317!RIV13-AV0-11320___
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http://linked.open...avai/riv/aktivita
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http://linked.open...avai/riv/aktivity
| - P(IAA100190901), Z(MSM0021620839)
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http://linked.open...iv/cisloPeriodika
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http://linked.open...vai/riv/dodaniDat
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http://linked.open...aciTvurceVysledku
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http://linked.open.../riv/druhVysledku
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http://linked.open...iv/duvernostUdaju
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http://linked.open...titaPredkladatele
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http://linked.open...dnocenehoVysledku
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http://linked.open...ai/riv/idVysledku
| - RIV/00216208:11320/12:10127317
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http://linked.open...riv/jazykVysledku
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http://linked.open.../riv/klicovaSlova
| - James's Compactness Theorem; measure of weak non-compactness; Banach space (en)
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http://linked.open.../riv/klicoveSlovo
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http://linked.open...odStatuVydavatele
| - GB - Spojené království Velké Británie a Severního Irska
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http://linked.open...ontrolniKodProRIV
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http://linked.open...i/riv/nazevZdroje
| - Proceedings of the Edinburgh Mathematical Society
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http://linked.open...in/vavai/riv/obor
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http://linked.open...ichTvurcuVysledku
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http://linked.open...cetTvurcuVysledku
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http://linked.open...vavai/riv/projekt
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http://linked.open...UplatneniVysledku
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http://linked.open...v/svazekPeriodika
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http://linked.open...iv/tvurceVysledku
| - Kalenda, Ondřej
- Spurný, Jiří
- Cascales, Bernardo
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http://linked.open...ain/vavai/riv/wos
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http://linked.open...n/vavai/riv/zamer
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issn
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number of pages
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http://bibframe.org/vocab/doi
| - 10.1017/S0013091510000842
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http://localhost/t...ganizacniJednotka
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