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Transactions of the American Mathematical Society

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

 

 

Complemented subspaces of products of Banach spaces


Authors: Paweł Domański and Augustyn Ortyński
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
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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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DOI: https://doi.org/10.1090/S0002-9947-1989-0937243-4
Keywords: Complemented subspaces, injective spaces, $ {L_1}$-preduals, the Dunford-Pettis property, products of Banach spaces, locally convex spaces
Article copyright: © Copyright 1989 American Mathematical Society