Historically, the idea of nonstandard analysis was to rigorously justify calcu- tions with in?nitesimal numbers. For example, formally, the chain rule of Leibniz calculus for the function F = f(g(x)) can be written as dF dF dg =, dx dg dx and for a formal proof, one may just divide numerator and denominator by the in?nitesimal small number dg. Nowadays, nonstandard analysis has gone far beyond the realm of in?nit- imals. In fact, it provides a machinery which enables one to describe explicitly mathematicalconcepts whichby standardmethods canonly be described impl- itly and in a cumbersome way. In the above example the standard notion of a limit is in a certain sense replaced by the nonstandard notion of an in?nit- imal. If one applies a similar approach to other objects than the real numbers (like topological spaces or Banach spaces etc. ), one has a tool which provides - plicit de?nitions for objects which can in principle not be described explicitly by standard methods. Examples of such objects are sets which are not Lebesgue measurable, orfunctionalswithcertainpropertieslikeso-calledHahn Banachl- its. Since it is possible in nonstandard analysis to simply calculate with such objects, one can obtain results about them which are extremely hard to obtain by standard methods. This book is an introduction to nonstandard analysis. In contrast to some other textbooks on this topic, it is not meant as an introduction to basic calculus by nonstandard analysis."
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Archimedean assume axiom of choice Banach-Mazur limit bijection binary relation cardinality Cauchy filter Cauchy sequence constant contains Conversely Corollary countable CT1R Dedekind complete defined deﬁnition denote elements equivalence class Exercise filter fin(*lR ﬁnite finite intersection property ﬁrst formula function f G 1R+ G CTN G inf(*lR G Noo G R+ h G Nqo Hausdorff space Hence holds induction inﬁnite infinitesimal internal definition principle internal entity internal function internal sets internal subsets interpretation map Lemma linear mon(x Moreover nonempty nonstandard analysis nonstandard embedding open sets particular permanence principle Proof Proposition prove pseudometric real numbers resp satisfies shortcut standard definition principle statement follows superstructure Theorem 7.1 topological space topological vector space transfer principle implies transitively bounded sentence true ultrafilter ultrapower uniform space uniform structure upper bound