Historically, mathematical analysis originated in17th century, accompanied by Newton and Leibniz invented calculus. The topics of mathematical analysis in 17 and 18 centuries, such as variational method, ordinary differential equation and partial differential equation, Fourier analysis and generating function, are basically developed in the application work. Calculus method successfully approximates discrete problems with continuous method.
In the whole18th century, the definition of function concept has become a controversial topic among mathematicians. /kloc-In the 9th century, Cauchy introduced the concept of Cauchy sequence and established calculus on a solid logical basis for the first time. He also founded the formal theory of complex analysis. Poisson, joseph liouville, Fourier and other mathematicians studied partial differential equations and harmonic analysis.
In the middle of that century, Riemann introduced his integral theory. In the last three decades of the19th century, the arithmeticization of Veiershtrass's analysis came into being. He thinks that geometric argument is misleading in nature and introduces the definition of limit (ε, δ). At this point, mathematicians began to worry that they assumed the existence of real continuum without proof. Dai Dejin used Dydykin division to construct real numbers. At about that time, the attempt to improve Riemann integral theorem also led to the study of the "size" of discontinuous sets of real number functions.
In addition, functions that are discontinuous everywhere, functions that are continuous everywhere but not differentiable, and space filling curves are also created. In this context, Jordan developed his measure theory, Cantor developed the present naive set theory, and Bell proved Bell's theorem. In the early 20th century, axiomatic set theory formalized calculus. Leberg solved the measure problem, and Hilbert introduced Hilbert space to solve the integral equation. The idea of normed vector space began to spread, and in the1920s, Barnach founded functional analysis.
Mathematical analysis is currently divided into the following sub-fields:
Real analysis is a rigorous study of the differential and integral forms of real functions. This includes the study of limit, power series and measure.
Functional analysis studies function space, and introduces concepts such as Banach space and Hilbert space.
Harmonic analysis involves Fourier series and its abstraction.
Complex analysis is the study of complex differentiable functions from complex plane to complex plane.