This book introduces Robinson's nonstandard analysis, an application of model theory in analysis. Unlike some texts, it does not attempt to teach elementary calculus on the basis of nonstandard analysis, but points to some applications in more advanced analysis. The contents proceed from a discussion of the preliminaries to Nonstandard Models; Nonstandard Real Analysis; Enlargements and Saturated Models; Functionals, Generalized Limits, and Additive Measures; and finally Nonstandard Topology and Functional Analysis. No background in model theory is required, although some familiarity with analysis, topology, or functional analysis is useful. This self-contained book can be understood after a basic calculus course.
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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