Synopsis
Nonstandard analysis was invented by Abraham Robinson in the 1960s as a way to rescue the naïve use of infinitesimal and infinite elements favored bymathematicians such as Leibniz and Euler before the advent of the rigorous methods introduced by Cauchy and Weierstrauß. Indeed, Robinson realized that the compactness theorem of first-order logic could be used to provide fields that “logically behaved" like the ordered real field while containing “ideal" elements such as infinitesimal and infinite elements. Since its origins, nonstandard analysis has become a powerful mathematical tool, not only for yielding easier definitions for standard concepts and providing slick proofs of well-known mathematical theorems, but for also providing mathematicians with amazing new tools to prove theorems, e.g. hyperfinite approximation. In addition, by providing useful mathematical heuristics a precise language to be discussed, many mathematical ideas have been elucidated greatly. In this treatise a wide spectrum of applications of nonstandard methods. In the first part of these notes, an explanation what a nonstandard extension is and it is used to reprove some basic facts from calculus. The nonstandard framework to handle more sophisticated mathematical situations and begin studying metric space topology. Then the realm of functional analysis is entered by discussing Banach and Hilbert spaces. Here, the first serious theorems are proven: the Spectral Theorem for compact Hermitian operators and the Bernstein Robinson Theorem on invariant subspaces; this latter theorem was the first major theorem whose first proof was nonstandard. Then a brief discussion of Loeb measure and using it to give a slick proof of an important combinatorial result, the Szemerédi Regularity Lemma.
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Nonstandard Analysis
Vendeur : Revaluation Books, Exeter, Royaume-Uni
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Paperback. Etat : Brand New. 155 pages. 9.00x6.00x0.39 inches. In Stock. N° de réf. du vendeur zk1088985017
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