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Lipschitz smooth points of convex functions and isomorphic characterizations of Hilbert spaces. (English) Zbl 0549.46025
Let X be a Banach space with dual \(X^*\). The following strengthening of the concept of Fréchet differentiability is introduced: A function \(\phi:X\to {\mathbb{R}}\) is called Lipschitz smooth at \(x\in X\) if there are \(c>0\), \(\delta >0\), and \(\xi\in X^*\) such that \(|\phi (x+h)-\phi(x)-<\xi,h>|\leq c\| h\|^ 2\) whenever \(h\in X\), \(\| h\| <\delta\). The X is called an LD-space if each convex continuous function on X is Lipschitz smooth on a dense subset of X. The aim of the paper is to built more or less succesfully a theory of LD-spaces analogous to that of Asplund spaces. It is shown, among other things, that the LD-spaces are separably determined and a ”Lipschitz” analogy of a theorem of Ekeland and Lebourg is proved. The introduced notion is used in characterizing Hilbert spaces: If X (or \(X^*)\) admits a smooth function with bounded nonempty support and locally Lipschitz derivative and \(X^*\) (resp. X) is an LD-space, then X is isomorphic to a Hilbert space.
Reviewer: M.Fabian

46G05 Derivatives of functions in infinite-dimensional spaces
46B20 Geometry and structure of normed linear spaces
46C99 Inner product spaces and their generalizations, Hilbert spaces
46A55 Convex sets in topological linear spaces; Choquet theory
46C20 Spaces with indefinite inner product (Kreĭn spaces, Pontryagin spaces, etc.)
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