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Convergence of sequences of tangentiable functions. (Convergence de suites de fonctions tangentiables.) (French) Zbl 1337.58006

Let (\(M,d)\), \((M',d')\) be metric spaces and \(f\), \(g:M\to M'\) two mappings. One says that \(f\), \(g\) are tangent at \(a\in M\), denoted \(f\asymp_a g\), if there exist a neighborhood \(V\) of \(a\) and a function \(c:V\to\mathbb R_+\) continuous at \(a\), with \(c(a) =0\), such that \(d'(f(x),g(x))\leq c(x)d(x,a)\) for all \(x\in V\). The function \(f\) is called tangentiable at \( a\) if there exists a locally Lipschitz function \(g\) tangent to \(f\) at \(a\). The relation \(\asymp_a\) is an equivalence relation in the set LL\(((M,a),M',a')\) of locally Lipschitz at \(a\) functions \(f:(M,a)\to (M',a')\) such that \(f(a)=a'\). One denotes by \(\mathbb J\mathrm{et}((M,a),(M',a'))\) the quotient LL\(((M,a),M',a')/\asymp_a\), whose elements are called linked metric jets from \((M,a)\) to \((M',a')\). The author in cooperation with E. Burroni developed in a series of papers a calculus of tangentiable functions, extending Ehresmann’s calculus of jets to the metric setting. A detailed presentation is given in [E. Burroni and J. Penon, “Elements for a metric tangential calculus”, Preprint, arXiv:0912.1012].
The present paper is concerned with convergence results for sequences of tangentiable functions.
As a sample we quote Theorem 1.1. One supposes that \(E,E'\) are normed spaces, \(U\subset E\) is convex, and \(f_n:U\to E'\) is a sequence of tangentiable functions. If
(i)
there exists \(a\in U\) such that the sequence \((f_n(a))\) is convergent in \(E'\),
(ii)
the sequence of tangents \((\mathrm{t}f_n(x))\) is uniformly convergent on \(U\) to a function \(\varphi:U\to\mathbb J\mathrm{et}((E,0),(E',0))\),
then the sequence \((f_n) \) converges pointwisely on \(U\) to a function \(f\) such that t\(f_x=\varphi(x)\) for every \(x\in U\).
The analogy with the corresponding result for sequences of differentiable functions is obvious.

MSC:

58C25 Differentiable maps on manifolds
58C20 Differentiation theory (Gateaux, Fréchet, etc.) on manifolds
58A20 Jets in global analysis
54E35 Metric spaces, metrizability
18D20 Enriched categories (over closed or monoidal categories)
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