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Topological properties of the set of norm-attaining linear functionals. (English) Zbl 0831.46009
Summary: If $$X$$ is a separable nonreflexive Banach space, then the set NA of all norm-attaining elements of $$X^*$$ is not a $$w^*$$-$$G_\delta$$ subset of $$X^*$$. However, if the norm of $$X$$ is locally uniformly rotund, then the set of norm-attaining elements of norm one is $$w^*$$-$$G_\delta$$. There exist separable spaces such that NA is a norm-Borel set of arbitrarily high class. If $$X$$ is separable and nonreflexive, there exists an equivalent Gâteaux-smooth norm on $$X$$ such that the set of all Gâteaux-derivatives is not norm-Borel.

##### MSC:
 46B20 Geometry and structure of normed linear spaces 46B03 Isomorphic theory (including renorming) of Banach spaces 03E15 Descriptive set theory
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