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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.
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
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