Real analysis is the part of mathematical analysis, as taught especially in undergraduate and graduate courses, that develops calculus rigorously over the real numbers and Euclidean spaces. An introductory course, sometimes called advanced calculus, covers limits, continuity, compactness, differentiation, integration and series, while advanced study extends into measure theory, Lebesgue integration and function spaces. The field was historically also called the theory of functions of a real variable, distinguishing it from the theory of a complex variable.
Facts
Central QuestionReal analysis asks how the operations of calculus, limits, continuity, differentiation and integration, can be given a rigorous logical foundation over the real numbers rather than resting on an unrigorous intuitive notion of infinitesimally small quantities. 2 Key DebateWhether infinitesimal quantities are a legitimate foundation for analysis was disputed for two centuries after Bishop Berkeley attacked their use as incoherent: the followers of Cantor, Dedekind and Weierstrass sought to eliminate infinitesimals in favor of the epsilon-delta definition of limit, while Hermann Cohen's Marburg school pursued a rigorous logic of infinitesimals instead, a dispute only settled in the 20th century by Abraham Robinson's non-standard analysis. 1 Classification
Pure or Applied Connections
Sources
1. Infinitesimal (Wikipedia)
History section, Berkeley and the Marburg school passageQuote, History section, Berkeley and the Marburg school passage
While the followers of Cantor, Dedekind, and Weierstrass sought to rid analysis of infinitesimals, and their philosophical allies like Bertrand Russell and Rudolf Carnap declared that infinitesimals are pseudoconcepts, Hermann Cohen and his Marburg school of neo-Kantianism sought to develop a working logic of infinitesimals.
2. Real Analysis (Wikipedia)
Wikimedia FoundationLead sectionQuote, Lead section
Real analysis is the part of mathematical analysis, especially as taught in undergraduate and graduate courses, that develops calculus rigorously over the real numbers and Euclidean spaces.
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