Abstract

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)

Highlights

  • The goal of this note is to introduce some Banach space theory in the definable context of o-minimal expansions of real closed fields

  • An important observation made during the course of their proof is that reparameterizations converge to reparameterizations having a sufficient degree of differentiability

  • This will furnish the substance of Remark 4.1, and provide some details of a theory which we believe to be interesting in its own right, laying the groundwork for what we hope will be a fruitful area of future analysis

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Summary

Introduction

The goal of this note is to introduce some Banach space theory in the definable context of o-minimal expansions of real closed fields The development of this theory was originally motivated by the search for the proof of the Reparameterization Theorem of Pila and Wilkie (Theorems 2.3 and 2.5 of [2]). An important observation made during the course of their proof is that reparameterizations converge to reparameterizations having a sufficient degree of differentiability This property is a consequence of the theory developed in this paper, in particular Corollary 3.7, a fact explained by Remark 4.1 of [2]. It is our aim in this note to provide an introductory exposition of this theory, outlining results which go beyond Remark 4.1.

Definitions
Function Spaces
Definable Convergence
Main Theorem
Further Results
Final remarks

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