Mathematics Department

Let (X , Γ) be a uniform space with its uniformity generated by a set of pseudo-metrics Γ. Let the symbol ≃ denote the usual infinitesimal relation on *X , and define a new infinitesimal relation ≈ on *X by writing x ≈ y whenever *ϱ (x, p ) ≃ *ϱ (y, p ) for each ϱ ∈ Γ and each p ∈ X . We call (X , Γ) an S-space if the relations ≃ and ≈ coincide on fin(*X ). S -spaces are interesting because their nonstandard hulls have representations within Nelson's internal set theory (IST, [5]). This was shown in [1], where it was also observed that the class of uniform spaces that have invariant nonstandard hulls is contained in the class of S -spaces. The question of whether there are S -spaces that do not have invariant nonstandard hulls was left open in [1]. In this note we show that when the uniformity of an S -space is given by a single pseudometric, the space has invariant nonstandard hulls. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

ABSTRACT Real Analysis Through Modern Infinitesimals provides a course in mathematical analysis based on Internal Set Theory (IST) introduced by Edward Nelson in 1977. After motivating IST through an ultrapower construction, the book provides a careful development of this theory representing each external class as a proper class. This foundational discussion, which is presented in the first two chapters, includes an account of the basic internal and external properties of the real number system as an entity within IST. In its remaining fourteen chapters, the book explores the perspective offered by IST as a wide range of real analysis topics are surveyed. The topics thus developed begin with those usually discussed in an advanced undergraduate analysis course and gradually move to topics that are suitable for more advanced readers. This book may be used for reference, self-study, and as a source for advanced undergraduate or graduate courses.

This book is unique in certain technical aspects of its approach, in its scale of coverage, and in its style of presentation --thoroughness, clarity, motivational material, philosophical undertone, and pedagogical considerations. The remarks, in the preface and the introduction, provide some evidence of these features.

Let ⟨X,Γ⟩ be a uniform space with its uniformity generated by a set of pseudo-metrics Γ. Let the symbol “ ≃ ” denote the usual infinitesimal relation on ∗X, and define a new infinitesimal relation “ ≈ ” by writing x ≈ y whenever ∗ρ(x,p) ≃ ∗ρ(y,p) for each ρ ∈ Γ and each p ∈ X. Let us call a uniform space ⟨X,Γ⟩ an S-space if the relations ≃ and ≈ coincide on fin(∗X). In [1], we showed that every S-space whose uniformity is generated by a single pseudometric has invariant nonstandard hulls. Here we extend that result to all uniform spaces and use it to explore further properties of S-spaces, which can now be recognized as uniform spaces that have invariant nonstandard hulls.

We introduce and study the notion of pseudo-uniform convergence which is a weaker variant of quasi-uniform convergence. Applications include the following nonstandard characterization of weak convergence. Let X X be an infinite set, B ( X ) B(X) the Banach space of all bounded real-valued functions on X , X, { f n : n ∈ N } \{f_{n}: n\in N\} a bounded sequence in B ( X ) , B(X), and f ∈ B ( X ) . f\in B(X). Then the sequence converges weakly to f f if and only if the convergence is pointwise on X X and, for each strictly increasing function σ : N → N \sigma :N\to N , each x ∈ ∗ X x\in ^{*}X , and each n ∈ ∗ N ∞ n\in ^{*}N_{\infty } , there is an unlimited m ≤ n m\leq n such that ∗ f ∗ σ ( m ) ( x ) ≃ ∗ f ( x ) ^{*}f_{ ^{*}\sigma (m)}(x) \simeq ^{*}f(x) .