Metric spaces and ideals of functions

Abstract

In each metric space (X, d) there is defined the space LipXof complex-valued, bounded, and uniformly Lipschitzian functions. In the algebra LipX, it is natural to ask for ideals closed in various notions of convergence, and also to identify the invertible elements. In particular, are the invertible elements exactly those with no zero inX? Wiener's Tauberian Theorem in Fourier analysis is the first and most remarkable example of this harmonious state of affairs. A moment's reflection confirms that, for the algebra LipX, this is true only for compact metric spacesX, the trivial examples in our investigation. We therefore introduce a type of convergence weaker than convergence in norm; it has already proved useful in some problems in descriptive set theory and reflects in a subtle way the metric properties ofX. A sequence (fn) in LipX converges stronglytog, writtens– limfn=g, if ∥fn∥≦Cin the Banach space LipXand limfn(x)=g(x)for each elementxofX. In Section 3 we explain how this is really a type of convergence in the dual space of a certain Banach space. This brings us to the edge of some recondite questions about iterated (or even transfinite) limits, and we have adhered to the notion of strong limits to avoid these questions. To illustrate the differences between these two approaches, we mention this problem: which maximal ideals of LipXare closed with respect to strong convergence of sequences? This isnotthe problem studied in Section 1.