Let be complete metric spaces and be Banach spaces. In the paper under review, the author considers spaces of functions from into (and from into ) determined by a generalized modulus of continuity. These spaces are called generalized Lipschitz spaces. This notion serves to unify the study of spaces of Lipschitz, little Lipschitz and uniformly continuous functions. The main purpose of this paper is to characterize all biseparating operators between generalized Lipschitz spaces. Two functions and defined on with values in are said to be disjoint if, for all , either or . A map between vector-valued function spaces is called separating if maps disjoint functions to disjoint functions. It is biseparating if is invertible and both and are separating.
A generalized Lipschitz space of real-valued functions is said to be Lipschitz normal if, for every pair of subsets of with , there exists , , such that on and on . A generalized Lipschitz space of -valued functions is called Lipschitz normal if is.
The main result of the paper is Theorem 16, which goes as follows. Suppose that and are generalized Lipschitz spaces which are Lipschitz normal. If is a linear biseparating map, then there exists a homeomorphism and, for each , a vector space isomorphism such that for all .
Also, the continuity properties of the family and of the operator with respect to suitable topologies are studied. Note that the functions in generalized Lipschitz spaces (as well as the metric spaces and ) may be unbounded. The usual compactification procedures for the spaces and are not required.