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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
##### MSC:
 46B10 Duality and reflexivity in normed spaces 54B20 Hyperspaces (general topology) 46B20 Geometry and structure of normed linear spaces
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