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On Asplund functions. (English) Zbl 1060.46505
The concept of Asplund function is defined as a continuous convex function $$f$$ on a Banach space $$X$$ such that every continuous convex function $$h\leq f$$ is Fréchet smooth on a residual set. The author gives five equivalent conditions for $$f$$ to be an Asplund function which are essentially in the spirit how an Asplund space can be equivalently introduced. To name at least one typical:
For each separable subspace $$Y\subset X$$, the set dom$$(f_{| Y})^*$$ is separable. In particular, $$X$$ is an Asplund space if and only if the norm is an Asplund function.
Further, the author proves that if $$f$$ is an Asplund function, then the restriction $$f_{| Y}$$ on any subspace $$Y$$ is Asplund as well as the quotient function $\hat {f}:X/Y \rightarrow \mathbb {R},\;\hat {f}(\hat {x})=\inf \{f(x+y)\mid y\in Y\}.$ However, it does not follow that $$f$$ is Asplund provided $$f_{| Y}$$ and $$\hat {f}$$ are.
In the end, a sufficient condition for $$f$$ to be an Asplund function is given by means of upper-semicontinuity of the subdifferential map.
##### MSC:
 46B20 Geometry and structure of normed linear spaces 46G99 Measures, integration, derivative, holomorphy (all involving infinite-dimensional spaces) 58C20 Differentiation theory (Gateaux, Fréchet, etc.) on manifolds
##### Keywords:
Fréchet differentiability; convex function; Asplund space
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