Publication Date:
2016-03-19
Description:
This paper deals with the Bishop–Phelps–Bollobás property ( BPBP ) on bounded closed convex subsets of a Banach space $X$ , not just on its closed unit ball $B_X$ . We prove that BPBP holds for bounded linear functionals on arbitrary bounded closed convex subsets of a real Banach space. We show that, for a Banach space $Y$ with property $(\beta ),$ the pair $(X,Y)$ has 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 a positive modulus of convexity, we show that the pair $(X,Y)$ has BPBP on $D$ for every Banach space $Y$ . We further obtain that, for an Asplund space $X$ and for a locally compact Hausdorff space $L$ , the pair $(X, C_0(L))$ has BPBP on every bounded closed absolutely convex subset $D$ of $X$ . Finally, we study the stability of BPBP on a bounded closed convex set for the $\ell _1$ -sum or $\ell _{\infty }$ -sum of a family of Banach spaces.
Print ISSN:
0024-6107
Electronic ISSN:
1469-7750
Topics:
Mathematics
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