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On operators which attain their norm

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Abstract

The following problem is considered. LetX andY be Banach spaces. Are those operators fromX toY which attain their norm on the unit cell ofX, norm dense in the space of all operators fromX toY? It is proved that this is always the case ifX is reflexive. In general the answer is negative and it depends on some convexity and smoothness properties of the unit cells inX andY. As an application a refinement of the Krein-Milman theorem and Mazur’s theorem concerning the density of smooth points, in the case of weakly compact sets in a separable space, is obtained.

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Authors and Affiliations

  1. Yale University, New Haven

    Joram Lindenstrauss

  2. University of Washington, Seattle

    Joram Lindenstrauss

Authors
  1. Joram Lindenstrauss
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Research supported by the National Science Foundation, U.S.A. (NSF-GP-378).

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Lindenstrauss, J. On operators which attain their norm. Israel J. Math. 1, 139–148 (1963). https://doi.org/10.1007/BF02759700

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  • DOI: https://doi.org/10.1007/BF02759700

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