Real Analysis Through Modern Infinitesimals
Cambridge University Press, 17 בפבר׳ 2011 - 565 עמודים
Real Analysis Through Modern Infinitesimals provides a course on 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 consequences of 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.
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Banach space called Cauchy sequence Cl(S closed condition continuous functions contradiction converges uniformly Corollary countable data are standard defined denoted derivative differentiable e m/(Z+ equation equivalent Exercise Let exists external characterization f is continuous finite Fix x e formula function f Hausdorff space Hence hypothesis implies induction inequality interval let f Let G Let the notation lim^oo limit point lZ[a mean value theorem metric space monotone n e Z+ nonempty set nonempty subset nonstandard normed vector space ns(X open set pointwise polynomial positive integer power series product topology Proof Assume Proof Let properties Prove the following Riemann integrable S-continuous standard set statement subspace symbols Theorem Let topological space topology totally bounded transfer axiom ul(Z+ uniform convergence uniformly continuous unlimited write