Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A quantitative version of James' compactness theorem   We introduce two measures of weak non-compactness $Ja_E$ and $Ja$ that
quantify, via distances, the idea of boundary behind James' compactness
theorem. These measures tell us, for a bounded subset $C$ of a Banach space $E$
and for given $x^*\in E^*$, how far from $E$ or $C$ one needs to go to find
$x^{**}\in \bar{C}^{w^*}\subset E^{**}$ with $x^{**}(x^*)=\sup x^* (C)$. A
quantitative version of James' compactness theorem is proved using $Ja_E$ and
$Ja$, and in particular it yields the following result: {\it Let $C$ be a
closed convex bounded subset of a Banach space $E$ and $r>0$. If there is an
element $x_0^{**}$ in $\bar{C}^{w^*}$ whose distance to $C$ is greater than
$r$, then there is $x^*\in E^*$ such that each $x^{**}\in\bar{C}^{w^*}$ at
which $\sup x^*(C)$ is attained has distance to $E$ greater than $r/2$.} 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.

21
The categories of this paper are: adap-org
13
The categories of this paper are: quant-ph
7
The categories of this paper are: hep-th
31
The categories of this paper are: ao-sci
32
The categories of this paper are: atom-ph
6
The categories of this paper are: astro-ph
14
The categories of this paper are: math-ph
9
The categories of this paper are: nlin
0
The categories of this paper are: hep-ph
15
The categories of this paper are: nucl-ex
16
The categories of this paper are: stat
5
The categories of this paper are: gr-qc
3
The categories of this paper are: physics
22
The categories of this paper are: chao-dyn
28
The categories of this paper are: alg-geom
30
The categories of this paper are: q-alg