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Abstract. For a large class of metric spaces with nice local structure, which includes Banach-Finsler manifolds and geodesic spaces of curvature bounded above, we give sufficient conditions for a local homeomorphism to be a covering projection. We first obtain a general condition in terms of a path continuation property. As a consequence, we deduce several conditions in terms of path-liftings involving a generalized derivative, and in particular we obtain an extension of Hadamard global inversion theorem in this context. Next we prove that, in the case of quasi-isometric mappings, some of these sufficient conditions are also necessary. Finally, we give some applications to the existence of global implicit functions.

In order to find metric spaces X for which the algebra Lip * (X) of bounded Lipschitz functions on X determines the Lipschitz structure of X, we introduce the class of small-determined spaces. We show that this class includes precompact and quasi-convex metric spaces. We obtain several metric characterizations of this property, as well as some other characterizations given in terms of the uniform approximation and the extension of uniformly continuous functions. In particular we show that X is small-determined if and only if every uniformly continuous real function on X can be uniformly approximated by Lipschitz functions.

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.

For a metric space X, we study the space D ∞ (X) of bounded functions on X whose pointwise Lipschitz constant is uniformly bounded. D ∞ (X) is compared with the space LIP ∞ (X) of bounded Lipschitz functions on X, in terms of different properties regarding the geometry of X. We also obtain a Banach-Stone theorem in this context. In the case of a metric measure space, we also compare D ∞ (X) with the Newtonian-Sobolev space N 1,∞ (X). In particular, if X supports a doubling measure and satisfies a local Poincaré inequality, we obtain that D ∞ (X) = N 1,∞ (X).

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We show that if X is a Banach space having an unconditional basis and a C p -smooth Lipschitz bump function, then for every C 1 -smooth function f from X into a Banach space Y , and for every continuous function ε : X → (0, ∞), there exists a C p -smooth function g : X → Y such that f − g ε and f − g ε.