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.


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


[1] DOI: 10.1112/jlms/s2-20.1.115 · Zbl 0431.46033 · doi:10.1112/jlms/s2-20.1.115
[2] Christensen, Measure theoretic zero sets in infinite dimensional spaces and applications to differentiability of Lipschitz mappings pp 10– (1973) · Zbl 0302.43001
[3] Bogachev, Math. Notes 36 pp 519– (1984) · Zbl 0576.28022 · doi:10.1007/BF01139552
[4] Aronszajn, Studia Math. 57 pp 147– (1976)
[5] Mankiewicz, Studia Math. 45 pp 15– (1973)
[6] Zajfček, Czechoslovak Math. J. 29 pp 340– (1979)
[7] Talagrand, C.R. Acad. Sci. Paris 288 pp 461– (1979)
[8] Phelps, Convex Functions, Monotone Operators and Differentiability: Lect. Notes in Math (1989) · Zbl 0658.46035 · doi:10.1007/978-3-662-21569-2
[9] Phelps, Pac. J. Math. 77 pp 523– (1978) · Zbl 0396.46041 · doi:10.2140/pjm.1978.77.523
[10] Zahorski, Bull. Soc. Math. France 74 pp 147– (1946)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.