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On the Bishop–Phelps–Bollobás property

Tom 119 / 2019

María D. Acosta Banach Center Publications 119 (2019), 13-32 MSC: Primary 46B04; Secondary 46B28. DOI: 10.4064/bc119-1

Streszczenie

The Bishop–Phelps Theorem states the denseness of the set of norm attaining functionals in the topological dual of any Banach space. Bollobás obtained a quantitative version of this result showing that a pair given by an element x in the unit sphere of a Banach space and a functional x^* in the unit sphere of the dual such that x^* (x) is close to 1 can be approximated in norm by another pair (y,y^*) satisfying the same conditions and also that y^* attains its norm at y. In 2008 the problem of obtaining versions of Bollobás result for operators was considered. In this survey we include most of the results on this topic to the present day.

Autorzy

  • María D. AcostaUniversidad de Granada
    Facultad de Ciencias
    Departamento de Análisis Matemático
    18071 Granada, Spain
    e-mail

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