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A characterization of Asplund spaces with the help of local \(\epsilon\)-supports of Ekeland and Lebourg. (English) Zbl 0577.46012

Let f be a function defined on a Banach space X with values in \((- \infty,+\infty]\) and let \(\epsilon >0\). According to I. Ekeland and G. Lebourg [Trans. Am. Math. Soc. 224, 193-216 (1977; Zbl 0313.46017)], f is said to be locally \(\epsilon\)-supported at \(x\in X\) if \(f(x)<+\infty\) and there exists a continuous linear functional \(\xi\) on X such that \(f(x+h)-f(x)\geq \xi (h)-\epsilon \| h\|\) for all \(h\in X\) with norm sufficiently small.
The aim of the note under review is to prove the following theorem: A Banach space X is an Asplund space (if and) only if for any \(\epsilon >0\) every lower semi-continuous function \(f: X\to (-\infty,+\infty]\) is locally \(\epsilon\)-supported at the points of a dense subset of \(\{x\in X:f(x)<+\infty \}\); thus answering affirmatively one of the questions posed by A. D. Ioffe in [Bull. Am. Math. Soc., New Ser. 10, 87-90 (1984; Zbl 0554.58008)]. This equivalence enables to extend a little results of N. V. Živkov from [Generic Gâteaux differentiability, directionally differentiable mappings, to appear in Rev. Roum. Math. Pures Appl.].
Thus, for instance, the following theorem holds: Let X be an Asplund space and \(T:X\to Y\) be a bounded linear operator with range dense in Y. Let \(f:Y\to (-\infty,+\infty)\) be a continuous mapping such that the limit \(\lim [f(y+th)-f(y)]/t\) for \(t\downarrow 0\) exists for each \(y\in Y\) and each \(h\in Y\). Then f is Gâteaux differentiable at the points of a \(G_{\delta}\) dense subset of Y. The proof of the announced characterization of Asplund spaces is based on a separable reduction.

MSC:

46B20 Geometry and structure of normed linear spaces
46G05 Derivatives of functions in infinite-dimensional spaces
58C20 Differentiation theory (Gateaux, Fréchet, etc.) on manifolds
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