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Article overview
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Bishop-Phelps-Bolloba's theorem on bounded closed convex sets | | Dong Hoon Cho
; Yun Sung Choi
; | | Date: |
10 Sep 2014 | | Abstract: | This paper deals with the emph{Bishop-Phelps-Bollob’as property}
(emph{BPBp} for short) on bounded closed convex subsets of a Banach space $X$,
not just on its closed unit ball $B_X$. We firstly prove that the emph{BPBp}
holds for bounded linear functionals on arbitrary bounded closed convex subsets
of a real Banach space. We show that for all finite dimensional Banach spaces
$X$ and $Y$ the pair $(X,Y)$ has the emph{BPBp} on every bounded closed convex
subset $D$ of $X$, and also that for a Banach space $Y$ with property $(eta)$
the pair $(X,Y)$ has the emph{BPBp} on every bounded closed absolutely convex
subset $D$ of an arbitrary Banach space $X$. For a bounded closed absorbing
convex subset $D$ of $X$ with positive modulus convexity we get that the pair
$(X,Y)$ has the emph{BPBp} on $D$ for every Banach space $Y$. We further
obtain that for an Asplund space $X$ and for a locally compact Hausdorff $L$,
the pair $(X, C_0(L))$ has the emph{BPBp} on every bounded closed absolutely
convex subset $D$ of $X$. Finally we study the stability of the emph{BPBp} on
a bounded closed convex set for the $ell_1$-sum or $ell_{infty}$-sum of a
family of Banach spaces. | | Source: | arXiv, 1409.3008 | | Services: | Forum | Review | PDF | Favorites | |
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