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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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On weakly countably determined Banach spaces
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by Sophocles Mercourakis PDF
Trans. Amer. Math. Soc. 300 (1987), 307-327 Request permission

Abstract:

For a topological space $X$, let ${C_1}(X)$ denote the Banach space of all bounded functions $f:X \to {\mathbf {R}}$ such that for every $\varepsilon > 0$ the set $\{ x \in X:|f(x)| \geqslant \varepsilon \}$ is closed and discrete in $X$, endowed with the supremum norm. The main theorem is the following: Let $L$ be a weakly countably determined subset of a Banach space; then there exist a subset $\Sigma ’$ of the Baire space $\Sigma$, a compact space $K$, and a bounded linear one-to-one operator $T:C(L) \to {C_1}(\Sigma ’ \times K)$ that is pointwise to pointwise continuous. In the case where $L$ is weakly analytic, $\Sigma ’$ can be replaced by $\Sigma$. This theorem is connected with the basic result of Amir-Lindenstrauss on WCG Banach spaces and has corresponding consequences such as: the representation of Gulko (resp. Talagrand) compact spaces as pointwise compact subsets of ${C_1}(\Sigma ’ \times K)$ (resp. ${C_1}(\Sigma \times K)$) (a compact space $\Omega$ is called Gulko or Talagrand compact if $C(\Omega )$ is WCD or a weakly $K$-analytic Banach space); the characterization of WCD (resp. weakly $K$-analytic) Banach spaces $E$, using one-to-one operators from ${E^{\ast }}$ into ${C_1}(\Sigma ’ \times K)$ (resp. ${C_1}(\Sigma \times K)$); and the existence of equivalent "good" norms on $E$ and ${E^{\ast }}$ simultaneously.
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Additional Information
  • © Copyright 1987 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 300 (1987), 307-327
  • MSC: Primary 46B20; Secondary 47B38, 54H05
  • DOI: https://doi.org/10.1090/S0002-9947-1987-0871678-1
  • MathSciNet review: 871678