"James's Compactness Theorem; measure of weak non-compactness; Banach space"@en . . "2" . "Kalenda, Ond\u0159ej" . "A QUANTITATIVE VERSION OF JAMES'S COMPACTNESS THEOREM"@en . "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 . . "0013-0915" . . "120541" . "[08320EB5BCE3]" . "A QUANTITATIVE VERSION OF JAMES'S COMPACTNESS THEOREM" . "RIV/00216208:11320/12:10127317" . . . "3"^^ . "A QUANTITATIVE VERSION OF JAMES'S COMPACTNESS THEOREM"@en . . . "2"^^ . . "Spurn\u00FD, Ji\u0159\u00ED" . "A QUANTITATIVE VERSION OF JAMES'S COMPACTNESS THEOREM" . . . "RIV/00216208:11320/12:10127317!RIV13-AV0-11320___" . "10.1017/S0013091510000842" . . "11320" . "GB - Spojen\u00E9 kr\u00E1lovstv\u00ED Velk\u00E9 Brit\u00E1nie a Severn\u00EDho Irska" . "000303129100006" . . "http://dx.doi.org/10.1017/S0013091510000842" . "P(IAA100190901), Z(MSM0021620839)" . . . . . . "Cascales, Bernardo" . "18"^^ . "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." . "55" . . "Proceedings of the Edinburgh Mathematical Society" . .