Convergence of continuous linear functionals and their level sets. (English) Zbl 0662.46015

Let \(X\) be a real Banach space with continuous dual \(X^*\). We characterize both norm and weak* convergence of sequences in \(X^*\) to a nonzero limit in terms of the convergence of the level sets of the linear functionals. When \(X\) is reflexive, norm convergence is equivalent to the Mosco convergence of level sets. Using this fact, we show that Mosco convergence of sequences of closed convex sets in a reflexive space may be properly stronger than pointwise convergence of the distance functions for the sets in the sequence.
Reviewer: G.Beer


46B10 Duality and reflexivity in normed linear and Banach spaces
54B20 Hyperspaces in general topology
46B20 Geometry and structure of normed linear spaces
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