Parties admissibles d’un espace de Banach. Applications. (English) Zbl 0544.46011
In this paper we consider the subsets A of a Banach space E such that the closure \(\tilde A\) of A in the bidual \(E^{**}\) equipped with the \(w^*\)-topology contains only ”regular” linear forms. It is shown that if a dual space \(E^*\) contains ”enough” subsets which satisfy this condition (so called ”admissible sets”) then the predual E of \(E^*\) is unique. This result gives all the previously known examples of uniqueness of the predual (and several others). Moreover, in many situations, we get that the uniqueness of the predual, in the isometric sense, is true for every equivalent dual norm on the space. Other applications are given: e.g. if E is a Banach space and if there exists an isometry S of \(E^{**}\) such that \(S^ 2=Id\) and \(Ker(S-Id)=E,\) then E is weakly sequentially complete. This result has been applied in ”Sous-espaces bien disposés de \(L^ 1\)” [to appear in Trans. Am. Math. Soc. (Zbl 0521.46012)].
MSC:
| 46B20 | Geometry and structure of normed linear spaces |
| 46B10 | Duality and reflexivity in normed linear and Banach spaces |
| 26A21 | Classification of real functions; Baire classification of sets and functions |
Keywords:
functions of the first Baire class; regular linear forms; admissible sets; uniqueness of the predual; equivalent dual normCitations:
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