Lipschitz-type functions on metric spaces

Abstract

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.

Two classes of metric spaces

“…Their results as well as the characterizations of small-determined contained in [2] are somehow similar. And, in fact, they can be considered as results given in an external way, since they need to embed the initial metric space into a family of different spaces constructed for each ε > 0.…”

“…Then, from Proposition 2.1 and Remark 2.2, it is only necessary to check the Lipschitz equivalence between d and ρ n , for every n ∈ N. Indeed, as we have said above the identity map id : (X, ρ n ) → (X, d) is always Lipschitz, while the other identity map id : (X, d) → (X, ρ n ) is Lipschitz in the small. So, we finish taking into account that every Lipschitz in the small function defined on a small-determined space is also Lipschitz (see [2]). …”

“…Nevertheless, as we can see in [2], small-determined metric spaces form a huge class of metric spaces containing for instance all the normed spaces and more generally all the length spaces. Moreover these spaces have good properties of approximation and extension of real-valued functions.…”

“…More precisely, one of them is related with the approximation and the extension of real-valued uniformly continuous functions (see [2]), whereas the other one is closer to boundedness and uniform bornologies (see [6]). More recently, in a paper devoted to the study of some special realcompactifications of metric spaces ( [4]), we have noticed that these worlds apparently far away can be connected.…”

“…Small-determined spaces. The class of small determined metric spaces, introduced in [2], is properly defined in terms of the so-called Lipschitz in the small functions. Recall that a function between metric spaces f : (X, d) → (Y, d ) is said to be Lipschitz in the small if there exist δ > 0 and K > 0 such that d (f (x), f (y)) ≤ K · d(x, y) whenever d(x, y) < δ.…”