Convergence results for function spaces over o-minimal structures

We begin the development of a theory of Banach spaces in the definable setting of o-minimal structures. We outline several results which develop the theory of compact embeddings for explicitly given function spaces. One aim is to explain the substantive underpinnings of an important observation used in the proof of the Reparameterization Theorem of Pila and Wilkie in (2). We place this observation in the broader context of our theory and demonstrate how it may be refined further. 2000 Mathematics Subject Classification 03C64 (primary); 46B99 (secondary)