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No infinite dimensional $ P$ space admits a Markuschevich basis

Author: William B. Johnson
Journal: Proc. Amer. Math. Soc. 26 (1970), 467-468
MSC: Primary 46.10
MathSciNet review: 0265916
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Abstract: Theorem. Let $ X$ be a Banach space. If $ X$ is a Grothendieck space and $ X$ admits a Markuschevich basis then $ X$ is reflexive. This theorem is used to prove the conjecture of J. A. Dyer [1] stated in the title.

References [Enhancements On Off] (What's this?)

  • [1] J. A. Dyer, Generalized bases in $ P$-spaces (preprint).
  • [2] A. Grothendieck, Sur les applications linéaires faiblement compactes d’espaces du type 𝐶(𝐾), Canadian J. Math. 5 (1953), 129–173 (French). MR 0058866
  • [3] Joram Lindenstrauss, Extension of compact operators, Mem. Amer. Math. Soc. No. 48 (1964), 112. MR 0179580

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Keywords: Markuschevich basis, complete biorthogonal systems, $ P$ spaces, injective Banach spaces, Grothendieck spaces
Article copyright: © Copyright 1970 American Mathematical Society

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