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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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Complemented subspaces of products of Banach spaces
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by Paweł Domański and Augustyn Ortyński PDF
Trans. Amer. Math. Soc. 316 (1989), 215-231 Request permission

Abstract:

It is proved that: (i) every complemented subspace in an infinite product of ${L_1}$-predual Banach spaces $\prod \nolimits _{i \in I} {{X_i}}$ is isomorphic to $Z \times {{\mathbf {K}}^\mathfrak {m}}$, where $\dim {\mathbf {K}} = 1,\;\mathfrak {m} \leqslant \operatorname {card} I$ and $Z$ is isomorphic to a complemented subspace of $\prod \nolimits _{i \in J} {{X_i},\;J \subseteq I,\;Z}$ contains an isomorphic cop[ill] of $c_0^{\operatorname {card} J}$; (ii) every injective lcs (in particular, Fréchet) is of the form $Z \times {{\mathbf {K}}^\mathfrak {m}},\;\dim {\mathbf {K}} = 1$, where $Z$ has a fundamental family of seminorms of the cardinality $\tau$ and $Z$ contains an isomorphic copy of $l_\infty ^\mathfrak {r}$ (this is a generalization of Lindenstrauss’ theorem on injective Banach spaces); (iii) whenever $X \simeq {l_p},\;1 \leqslant p \leqslant \infty$, or $X \simeq {c_0}$, then every complemented subspace in a power ${X^\mathfrak {m}}$ ($\mathfrak {m}$ is an arbitrary cardinal number) is isomorphic to ${X^\mathfrak {r}} \times {{\mathbf {K}}^\mathfrak {s}},\;\mathfrak {r} + s \leqslant \mathfrak {m}$ (a generalization of the results due to Lindenstrauss and Pełczyński for $\mathfrak {m} = 1$).
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Additional Information
  • © Copyright 1989 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 316 (1989), 215-231
  • MSC: Primary 46A05; Secondary 46A45, 46B99, 46M99
  • DOI: https://doi.org/10.1090/S0002-9947-1989-0937243-4
  • MathSciNet review: 937243