Universally Lusin-measurable and Baire-$1$ projections
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- by Elias Saab PDF
- Proc. Amer. Math. Soc. 78 (1980), 514-518 Request permission
Abstract:
It is obvious that a dual Banach space ${E^\ast }$ is reflexive if and only if the natural projection P from ${E^{ \ast \ast \ast }}$ to ${E^\ast }$ is ${\text {weak}^\ast }$ to weak continuous. In this paper it is proved that the next best condition on P, namely that P is ${\text {weak}^\ast }$ to weak universally Lusin-measurable is necessary and sufficient for ${E^\ast }$ to have the Radon-Nikodým property. In addition we prove that if E is any Banach space that is complemented in its second dual by a ${\text {weak}^\ast }$ to weak Baire-1 projection, then E has the Radon-Nikodým property. We also prove that if E is a Banach space that is complemented in its second dual ${E^{ \ast \ast }}$ by a projection $P:{E^{\ast \ast }} \to E$ with $F = {P^{ - 1}}(0)$ weakly K-analytic; then saying that ${E^{ \ast \ast }}$ has the Radon-Nikodým property is equivalent to saying that P is ${\text {weak}^\ast }$ to weak universally Lusin-measurable.References
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Additional Information
- © Copyright 1980 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 78 (1980), 514-518
- MSC: Primary 46B22
- DOI: https://doi.org/10.1090/S0002-9939-1980-0556623-X
- MathSciNet review: 556623