an:03875985
Zbl 0549.46025
Fabian, Mari??n
Lipschitz smooth points of convex functions and isomorphic characterizations of Hilbert spaces
EN
Proc. Lond. Math. Soc., III. Ser. 51, 113-126 (1985).
00149193
1985
j
46G05 46B20 46C99 46A55 46C20
Fr??chet differentiability; Lipschitz smooth; LD-space; convex continuous function; Asplund spaces; characterizing Hilbert spaces; locally Lipschitz derivative
Let X be a Banach space with dual \(X^*\). The following strengthening of the concept of Fr??chet 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.
M.Fabian