
doi: 10.1007/bf02390820
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 \(\ell_2\).)
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
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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