We study the invertibility nonsmooth maps between infinite-dimensional Banach spaces. To this end, we introduce an analogue of the notion of pseudo-Jacobian matrix of Jeyakumar and Luc in this infinite-dimensional setting. Using this, we obtain several inversion results. In particular, we give a version of the classical Hadamard integral condition for global invertibility in this context.

We study the global inversion of a continuous nonsmooth mapping f : R n → R n , which may be non-locally Lipschitz. To this end, we use the notion of pseudo-Jacobian map associated to f , introduced by Jeyakumar and Luc, and we consider a related index of regularity for f . We obtain a characterization of global inversion in terms of its index of regularity. Furthermore, we prove that the Hadamard integral condition has a natural counterpart in this setting, providing a sufficient condition for global invertibility.

It is well known that every bounded below and non increasing sequence in the real line converges. We give a version of this result valid in Banach spaces with the Radon-Nikodym property, thus extending a former result of A. Procházka. Introduction.Our purpose is to state an analogue of the fact that every bounded below and non increasing sequence in the real line R converges in the framework of a Banach space X. This is not clear, even whenever X = R 2 . However, we shall see that it is indeed possible in Banach spaces with the Radon-Nikodym property.Definition 1.1. Let X be a Banach space. We say that X has the Radon-Nikodym property if, for every non empty closed convex bounded subset C of X and every η > 0, there exists g in the unit sphere of the dual of X and c ∈ R such that {x ∈ C; g(x) < c} is non empty and has diameter less than η.Every reflexive Banach space has the Radon-Nikodym property, but L 1 ([0, 1]) and C(K) spaces whenever K is an infinite compact space fail this poperty. Moreover, if Y is a subspace of a Banach space with the Radon-Nikodym property, then Y has the Radon-Nikodym property. The Radon-Nikodym property can be characterized in many ways, see [1], [2] and [5].Before stating our main result, we need some notations. If X is a real Banach space, S X stands for its unit sphere and S X * for the unit sphere of its dual. For f ∈ X * and r > 0 we denote B(f, r) = {g ∈ X * : f − g r} and B(f, r) = {g ∈ X * : f − g < r} the closed and open ball centered at f and of radius r respectively. Let us recall that whenever X is a Banach space, g ∈ X * and c ∈ R, we denote {g c} the closed half space {u ∈ X; g(u) c} and {g < c} the open half space {u ∈ X; g(u) < c}. If C is a non empty convex subset of X, the set C ∩ {g c} is called a closed slice of C and D ∩ {g < c} an open slice of C. If x ∈ X and f ∈ X * , we shall use both notations y(x) and f, x for the evaluation of f at x. Theorem 1.2. Let X be a Banach space with the Radon-Nikodym property. Let f ∈ S X * and ε ∈ (0, 1) be fixed. There exists a function t : X → S X * ∩ B(f, ε) such

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Our aim in this paper is to study the global invertibility of a locally Lipschitz map f : X → Y between (possibly infinite-dimensional) Finsler manifolds, stressing the connections with covering properties and metric regularity of f . To this end, we introduce a natural notion of pseudo-Jacobian Jf in this setting, as is a kind of set-valued differential object associated to f . By means of a suitable index, we study the relations between properties of pseudo-Jacobian Jf and local metric properties of the map f , which lead to conditions for f to be a covering map, and for f to be globally invertible. In particular, we obtain a version of Hadamard integral condition in this context.