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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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