Editing Real analysis (section)

== Generalizations and related areas of mathematics ==
Various ideas from real analysis can be generalized from the real line to broader or more abstract contexts.  These generalizations link real analysis to other disciplines and subdisciplines. For instance, generalization of ideas like continuous functions and compactness from real analysis to [[metric space]]s and [[topological space]]s connects real analysis to the field of [[general topology]], while generalization of finite-dimensional Euclidean spaces to infinite-dimensional analogs led to the concepts of [[Banach space]]s and [[Hilbert space]]s and, more generally to [[functional analysis]].  [[Georg Cantor]]'s investigation of sets and sequence of real numbers, mappings between them, and the foundational issues of real analysis gave birth to [[naive set theory]].  The study of issues of [[Limit (mathematics)|convergence]] for sequences of functions eventually gave rise to [[Fourier analysis]] as a subdiscipline of mathematical analysis.   Investigation of the consequences of generalizing differentiability from functions of a real variable to ones of a complex variable gave rise to the concept of [[holomorphic function]]s and the inception of [[complex analysis]] as another distinct subdiscipline of analysis.  On the other hand, the generalization of integration from the Riemann sense to that of Lebesgue led to the formulation of the concept of abstract [[measure space]]s, a fundamental concept in [[measure theory]].  Finally, the generalization of integration from the real line to curves and surfaces in higher dimensional space brought about the study of [[vector calculus]], whose further generalization and formalization played an important role in the evolution of the concepts of [[differential form]]s and [[Differentiable manifold|smooth (differentiable) manifolds]] in [[differential geometry]] and other closely related areas of [[geometry]] and [[topology]].