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A class of null sets associated with convex functions on Banach spaces. (English) Zbl 0724.46017

Let E be a real Banach space, f: \(E\to {\mathbb{R}}\) be a continuous convex function. Denote by \({\mathcal N}(f)\) the set of all points \(x\in E\) where f fails to be Gateaux differentiable. A subset \(B\subset E\) is called null set if \(B\subset {\mathcal N}(f)\) for some f. It is proved that a countable union of null sets is again a null set.
The relation of this notion to other notions of null sets is discussed.

MSC:

46B20 Geometry and structure of normed linear spaces
46G05 Derivatives of functions in infinite-dimensional spaces
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