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On strong continuity of derivatives of mappings. (English) Zbl 0633.46044

Let \({\mathcal F}\) be a family of mappings defined on an open set (or a weak open set) in a locally convex topological vector space with values in another locally convex space, and denote \({\mathcal F}'\) the set of derivatives of \(f\in {\mathcal F}\), if it is differentiable in the sense of Gâteaux (or Fréchet).
The author investigates various relations among the following properties of \({\mathcal F}\) and \({\mathcal F}':\) weak and strong equicontinuity, uniformly weak (strong) equicontinuity, collective precompactness, Gâteaux and Fréchet differentiability at a point or uniformly on the domain, equidifferentiability in the sense of Gâteaux or Fréchet (uniformly on the domain, and (weak)pseudouniformly), and other relating properties.
This article is a continuation of an author’s paper: Commentat. Math. Univ. Carolinae 17, 7-30 (1976; Zbl 0321.58008). The author emphasizes that the results obtained here are new even in the case of single mappings and that some theorems concerning strong equicontinuity of \({\mathcal F}'\) are valid without any restriction on the spaces.
Reviewer: K.Furutani

MSC:

46G05 Derivatives of functions in infinite-dimensional spaces
58C20 Differentiation theory (Gateaux, Fréchet, etc.) on manifolds

Citations:

Zbl 0321.58008
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