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
Includes
Source Anderson's Theorem (Wikipedia)
Source Bernstein's Theorem on Monotone Functions (Wikipedia)
Source Cousin's Theorem (Wikipedia)
Source Ernst Leonard Lindelöf (Wikipedia)
Source Fubini's Theorem on Differentiation (Wikipedia)
Source Hardy-Littlewood Maximal Function (Wikipedia)
Source Interior Extremum Theorem (Wikipedia)
Source Kolmogorov-Arnold Representation Theorem (Wikipedia)
Source Lebesgue Differentiation Theorem (Wikipedia)
Source Projectively Extended Real Line (Wikipedia)
Source Riesz-Fischer theorem (Wikipedia)
Source Vitali convergence theorem (Wikipedia)
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.
Vitali convergence theorem (Wikipedia)
Includes: Vitali Convergence Theorem, Lead sentenceQuote, Includes: Vitali Convergence Theorem, Lead sentence
In real analysis and measure theory, the Vitali convergence theorem, named after the Italian mathematician Giuseppe Vitali, is a g
Anderson's Theorem (Wikipedia)
Includes: Anderson's Theorem, Lead sentenceQuote, Includes: Anderson's Theorem, Lead sentence
In mathematics, Anderson's theorem is a result in real analysis and geometry which says that the integral of an integrable, symmet
Bernstein's Theorem on Monotone Functions (Wikipedia)
Includes: Bernstein's Theorem on Monotone Functions, Lead sentenceQuote, Includes: Bernstein's Theorem on Monotone Functions, Lead sentence
In real analysis, a branch of mathematics, Bernstein's theorem, named after Sergei Bernstein, states that every real-valued functi
Cousin's Theorem (Wikipedia)
Includes: Cousin's Theorem, Lead sentenceQuote, Includes: Cousin's Theorem, Lead sentence
In real analysis, a branch of mathematics, Cousin's theorem states that: If for every point of a closed region (in modern terms, "
Interior Extremum Theorem (Wikipedia)
Includes: Interior Extremum Theorem, Lead sentenceQuote, Includes: Interior Extremum Theorem, Lead sentence
In calculus and real analysis, the interior extremum theorem states that any local extremum of a real function at which it is diff
Fubini's Theorem on Differentiation (Wikipedia)
Includes: Fubini's Theorem on Differentiation, Lead sentenceQuote, Includes: Fubini's Theorem on Differentiation, Lead sentence
tiation, named after Guido Fubini, is a result in real analysis concerning the differentiation of series of monotonic functions.
Projectively Extended Real Line (Wikipedia)
Includes: Projectively Extended Real Line, Lead sentenceQuote, Includes: Projectively Extended Real Line, Lead sentence
In real analysis, the projectively extended real line (also called the one-point compactification of the real line), is the extens
Kolmogorov-Arnold Representation Theorem (Wikipedia)
Includes: Kolmogorov-Arnold Representation Theorem, Lead sentenceView the Source Hardy-Littlewood Maximal Function (Wikipedia)
Includes: Hardy-Littlewood Maximal Function, Lead sentenceQuote, Includes: Hardy-Littlewood Maximal Function, Lead sentence
or M is a significant non-linear operator used in real analysis and harmonic analysis.
Lebesgue Differentiation Theorem (Wikipedia)
Wikimedia FoundationIncludes: Lebesgue Differentiation Theorem, Lead sentenceQuote, Includes: Lebesgue Differentiation Theorem, Lead sentence
Lebesgue differentiation theorem is a theorem of real analysis, which states that for almost every point, the value of an integra
Riesz-Fischer theorem (Wikipedia)
Ernst Leonard Lindelöf (Wikipedia)
Includes: Ernst Leonard Lindelof, Lead paragraph [in-branch]Quote, Includes: Ernst Leonard Lindelof, Lead paragraph [in-branch]
real analysis
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