On a three-space property for Lindelöf Sigma-spaces, (WCG)-spaces and the Sobczyk property

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 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 three-space property. (ii) The Lindelöf Σ-property is not a three-space 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 infinite-dimensional 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 non-Baire hyperplane. This partially answers a question of Saxon concerning Arias de Reyna-Valdivia-Saxon 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~three-space property. In the last part we recall some facts (partially with a simpler presentation) concerning K-analytic, Lindelöf ΣΣ and analytic locally convex spaces. Additionally, a few remarks concerning weakly K-analytic spaces are include