Abstract:

[EN] Corson's example shows that there exists a Banach space EE which is not weakly normal but EE contains a closed subspace isomorphic to the Banach space C[0,1]C[0,1] and such that the quotient space E/C[0,1]E/C[0,1] is ...[+]
[EN] Corson's example shows that there exists a Banach space EE which is not weakly normal but EE contains a closed subspace isomorphic to the Banach space C[0,1]C[0,1] and such that the quotient space E/C[0,1]E/C[0,1] is isomorphic to the weakly compactly generated Banach space c0[0,1]c0[0,1]. This applies to show the following two results: (i) The Lindelöf property is not a threespace property. (ii) The Lindelöf Σproperty is not a threespace property. In this note using the lifting property developed by Susanne Dierolf we present a very simple argument providing also (ii), see Theorem 1. This argument used in the proof applies also to show that under Continuum Hypothesis every infinitedimensional topological vector space EE which contains a dense hyperplane admits a stronger vector topology υυ with the same topological dual and such that (E,υ)(E,υ) contains a dense nonBaire hyperplane. This partially answers a question of Saxon concerning Arias de ReynaValdiviaSaxon theorem. A Banach space EE has the Sobczyk Property if it contains an isomorphic copy of c0c0 and every such a copy is complemented in EE. The classical Sobczyk's theorem says that every separable Banach space has this property. We give an example of a C(K)C(K)space EE and its subspace YY isometric to c0c0 such that E/YE/Y is isomorphic to c0(Γ)c0(Γ), with card(Γ)=2ℵ0card(Γ)=2ℵ0, yet YY is uncomplemented in EE. This complements Corson's example and shows that the Sobczyk Property (as well as the (WCG)property, and the Separable Complementation Property) is not a~threespace property. In the last part we recall some facts (partially with a simpler presentation) concerning Kanalytic, Lindelöf ΣΣ and analytic locally convex spaces. Additionally, a few remarks concerning weakly Kanalytic spaces are include
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Thanks:

The first author has been partially supported by MEC and FEDER Project MTM200803211. The research for the second named author was (partially) supported by Ministry of Science and Higher Education, Poland, grant no. NN201 ...[+]
The first author has been partially supported by MEC and FEDER Project MTM200803211. The research for the second named author was (partially) supported by Ministry of Science and Higher Education, Poland, grant no. NN201 2740 33, and for the second and third named author by the project MTM2008  01502 of the Spanish Ministry of Science and Innovation.
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