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On nonseparable Banach spaces


Author: Spiros A. Argyros
Journal: Trans. Amer. Math. Soc. 270 (1982), 193-216
MSC: Primary 46B20; Secondary 03E35, 03E50
DOI: https://doi.org/10.1090/S0002-9947-1982-0642338-2
MathSciNet review: 642338
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Abstract: Combining combinatorial methods from set theory with the functional structure of certain Banach spaces we get some results on the isomorphic structure of nonseparable Banach spaces. The conclusions of the paper, in conjunction with already known results, give complete answers to problems of the theory of Banach spaces. An interesting point here is that some questions of Banach spaces theory are independent of Z.F.C. So, for example, the answer to a conjecture of Pełczynski that states that the isomorphic embeddability of $ {L^1}{\{ - 1,\,1\} ^\alpha }$ into $ {X^{\ast}}$ implies, for any infinite cardinal $ \alpha $, the isomorphic embedding of $ l_\alpha ^1$ into $ X$, gets the following form:

if $ \alpha = \omega $, has been proved from Pełczynski;

if $ \alpha > {\omega ^ + }$, the proof is given in this paper;

if $ \alpha = {\omega ^ + }$, in $ {\text{Z}}{\text{.F}}{\text{.C}}{\text{.}} + {\text{C}}{\text{.H}}{\text{.}}$, an example discovered by Haydon gives a negative answer;

if $ \alpha = {\omega ^ + }$, in $ {\text{Z}}{\text{.F}}{\text{.C}}{\text{.}} + \urcorner {\text{C}}{\text{.H}}{\text{.}} + {\text{M}}{\text{.A}}{\text{.}}$, is also proved in this paper.


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DOI: https://doi.org/10.1090/S0002-9947-1982-0642338-2
Article copyright: © Copyright 1982 American Mathematical Society

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