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Banach spaces not containing l1

Banach spaces not containing \(\ell_{1}\)
Authors: Mujica, Jorge;

Banach spaces not containing l1

Abstract

The author presents an adjustment and extension of the proof of a 1977-result of J. Hagler and W. Johnson giving a sufficient condition for a real Banach space to contain a copy of \(\ell_1\). The famous Josefson-Nissenzweig theorem follows quite easily from the real Hagler-Johnson theorem. This clearly indicates that it is a central result in Banach space theory and, being such a structure theorem, it is important to know that it is valid also in the complex case. Having a complex version of the Hagler-Johnson theorem gives the possibility of a direct combination with the Rosenthal-Dor theorem. The author of this paper does so, and obtains the following result: Let \(E\) be a Banach space with no copy of \(\ell_1\). Then each infinite-dimensional subspace of \(E^{\prime}\) contains a normalized weak-star null sequence. Further combination with the Rosenthal-Dor theorem gives a beautiful link between the Dunford-Pettis property and containment of \(\ell_1\): Whenever \(E\) or \(E^{\prime}\) contains a subspace with the Dunford-Pettis property, then \(E\) contains a copy of \(\ell_1\). The author ends his paper by proving that if \(E^{\prime}\) contains a copy of \(\ell_1\), then \(E\) has a separable quotient (isomorphic to either \(c_0\) or \(\ell_2\).)

Related Organizations
Keywords

Josefsson-Nissenzweig theorem, copy of \(\ell_1\), Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces, Hagler-Johnson theorem, Classical Banach spaces in the general theory

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
1
Average
Average
Average
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gold