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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Twisted sums of Banach and nuclear spaces


Author: Paweł Domański
Journal: Proc. Amer. Math. Soc. 97 (1986), 237-243
MSC: Primary 46A22; Secondary 46A12, 46M10
DOI: https://doi.org/10.1090/S0002-9939-1986-0835872-2
MathSciNet review: 835872
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Abstract: A twisted sum of (topological vector) spaces $ Y$ and $ Z$ is a space $ X$ with a subspace $ {Y_1}$ isomorphic to $ Y$ for which $ X/{Y_1}$ is isomorphic to $ Z$. It splits if $ {Y_1}$ is complemented. It is proved that every twisted sum of a Banach space $ Y$ and a nuclear space $ Z$ splits. Köthe sequence spaces $ Z$ for which this holds are characterized. Every locally convex twisted sum of a nuclear Fréchet space $ Y$ and a Banach space $ Z$ splits too. If $ Z$ is superreflexive, then the local convexity assumption on the twisted sum may be omitted. Other results of this kind on Köthe sequence spaces are obtained.


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DOI: https://doi.org/10.1090/S0002-9939-1986-0835872-2
Keywords: Twisted sum, nuclear space, Fréchet space, Banach space, complemented subspace, lifting, Köthe sequence space, short exact sequence
Article copyright: © Copyright 1986 American Mathematical Society