realanalyse

Realanalyse, or real analysis, is the branch of mathematical analysis that studies real numbers, real-valued functions, and the limiting processes that arise in calculus. It provides a rigorous foundation for concepts such as limits, continuity, differentiation, integration, and infinite series, and it often emphasizes generalizations to functions of several real variables, metric spaces, and measures.

Core topics include sequences and series, convergence notions, continuity, differentiation, and the various modes of integration (Riemann and Lebesgue). The subject also investigates the structure of function spaces (for example C[a,b], Lp spaces) and the role of topology in analysis. Fundamental results include the Bolzano–Weierstrass theorem, the Heine–Borel theorem, monotone convergence, dominated convergence, and the fundamental theorem of calculus, which links differentiation and integration. Real analysis also covers completeness of the real numbers and properties of compactness, uniform convergence, and interchanging limits and integrals or derivatives under suitable conditions.

Historical note: Real analysis arose from the rigorization of calculus in the 19th century with mathematicians such as Bolzano, Cauchy, Weierstrass, and Riemann, and was extended by Lebesgue’s measure theory in the early 20th century.

Relation to related fields: Real analysis forms the foundation for more advanced analysis and functional analysis; its methods are central to probability theory, mathematical physics, and numerical analysis, among other areas.