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
DOI:
https://doi.org/10.1090/S0002-9939-1970-0265916-9
MathSciNet review:
0265916
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Abstract | References | Similar Articles | Additional Information
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.
-
J. A. Dyer, Generalized bases in $P$-spaces (preprint).
- A. Grothendieck, Sur les applications linéaires faiblement compactes d’espaces du type $C(K)$, Canad. J. Math. 5 (1953), 129–173 (French). MR 58866, DOI https://doi.org/10.4153/cjm-1953-017-4
- Joram Lindenstrauss, Extension of compact operators, Mem. Amer. Math. Soc. 48 (1964), 112. MR 179580
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Keywords:
Markuschevich basis,
complete biorthogonal systems,
<IMG WIDTH="21" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$P$"> spaces,
injective Banach spaces,
Grothendieck spaces
Article copyright:
© Copyright 1970
American Mathematical Society