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Operators attaining their norms, and the geometry of the unit sphere of a Banach space. (English. Russian original) Zbl 0757.46025
Sov. Math., Dokl. 42, No. 2, 532-534 (1991); translation from Dokl. Akad. Nauk SSSR 314, No. 4, 777-779 (1990).
The authors continue the investigation of the problem of the density of the norm-attaining operators in the space of all operators between Banach spaces, in the spirit of the results of Bishop-Phelps and Lindenstrauss. It is shown that in some sense “almost all” spaces $$X$$ admit an equivalent norm for which the norm-attaining operators from $$X$$ into an arbitrary space $$Y$$ are dense in the space of all operators from $$X$$ to $$Y$$. The sufficient condition on $$X$$ is that a “long biorthogonal system” should exist.
Reviewer: H.König (Kiel)

##### MSC:
 46B20 Geometry and structure of normed linear spaces 47A30 Norms (inequalities, more than one norm, etc.) of linear operators