Banach spaces with the condition of Mazur. (English) Zbl 0658.46010
A real Banach space X is said to be d-complete if every element of its second dual \(X^{**}\) that is sequentially weak* continuous is weak* continuous; this is called Mazur’s condition [J. Diestel, Lecture Notes Math. 541, 211-227 (1976; Zbl 0358.46030)] and has application in measure theory.
The author shows that X is d-complete if and only if X is complete with respect to the locally convex topology generated by all weak* compact and metrizable convex subsets of the dual \(X^*\). The property is naturally related to Grothendieck spaces and to results of G. A. Edgar [Indiana Univ. Math. J. 28, No.4, 559-579 (1979; Zbl 0418.46034)] on angelic spaces. Many sequence spaces are shown to be d-complete. A characterization of d-complete AL-spaces is given. Amongst those spaces of type C(K) where K is a compact topological space, d-complete spaces are discussed. The permanence property of d-completeness with respect to inductive and projective tensor products is investigated.
The author shows that X is d-complete if and only if X is complete with respect to the locally convex topology generated by all weak* compact and metrizable convex subsets of the dual \(X^*\). The property is naturally related to Grothendieck spaces and to results of G. A. Edgar [Indiana Univ. Math. J. 28, No.4, 559-579 (1979; Zbl 0418.46034)] on angelic spaces. Many sequence spaces are shown to be d-complete. A characterization of d-complete AL-spaces is given. Amongst those spaces of type C(K) where K is a compact topological space, d-complete spaces are discussed. The permanence property of d-completeness with respect to inductive and projective tensor products is investigated.
MSC:
| 46B20 | Geometry and structure of normed linear spaces |
| 46G12 | Measures and integration on abstract linear spaces |
| 46M05 | Tensor products in functional analysis |
| 46A50 | Compactness in topological linear spaces; angelic spaces, etc. |
Keywords:
d-complete; second dual; sequentially weak* continuous; Mazur’s condition; Grothendieck spaces; angelic spaces; d-complete AL-spaces; permanence property of d-completeness with respect to inductive and projective tensor productsReferences:
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