Fabian, M.; Zhivkov, N. V. A characterization of Asplund spaces with the help of local \(\epsilon\)-supports of Ekeland and Lebourg. (English) Zbl 0577.46012 C. R. Acad. Bulg. Sci. 38, 671-674 (1985). 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. Cited in 2 ReviewsCited in 13 Documents 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 Keywords:generic Gâteaux differentiability; locally \(\epsilon\)-supported; Asplund space; lower semi-continuous function Citations:Zbl 0313.46017; Zbl 0554.58008 PDFBibTeX XMLCite \textit{M. Fabian} and \textit{N. V. Zhivkov}, C. R. Acad. Bulg. Sci. 38, 671--674 (1985; Zbl 0577.46012)