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Article overview
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The Bishop-Phelps-Bollob'as version of Lindenstrauss properties A and B | Richard Aron
; Yun Sung Choi
; Sun Kwang Kim
; Han Ju Lee
; Miguel Martin
; | Date: |
28 May 2013 | Abstract: | We study a Bishop-Phelps-Bollob’as version of Lindenstrauss properties A and
B. For domain spaces, we study Banach spaces $X$ such that $(X,Y)$ has the
Bishop-Phelps-Bollob’as property (BPBp) for every Banach space $Y$. We show
that in this case, there exists a universal function $eta_X(eps)$ such that
for every $Y$, the pair $(X,Y)$ has the BPBp with this function. This allows us
to prove some necessary isometric conditions for $X$ to have the property. We
also prove that if $X$ has this property in every equivalent norm, then $X$ is
one-dimensional. For range spaces, we study Banach spaces $Y$ such that $(X,Y)$
has the Bishop-Phelps-Bollob’as property for every Banach space $X$. In this
case, we show that there is a universal function $eta_Y(eps)$ such that for
every $X$, the pair $(X,Y)$ has the BPBp with this function. This implies that
this property of $Y$ is strictly stronger than Lindenstrauss property B. The
main tool to get these results is the study of the Bishop-Phelps-Bollob’as
property for $c_0$-, $ell_1$- and $ell_infty$-sums of Banach spaces. | Source: | arXiv, 1305.6420 | Services: | Forum | Review | PDF | Favorites |
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