![]() ![]() ![]() The idea of normed vector space was in the air, and in the 1920s Banach created functional analysis. Lebesgue solved the problem of measure, and Hilbert introduced Hilbert spaces to solve integral equations. In the early 20th century, calculus was formalized using an axiomatic set theory. In this context, Jordandeveloped his theory of measure, Cantor developed what is now called naive set theory, and Baire proved the Baire category theorem. Around that time, the attempts to refine the theorems of Riemann integration led to the study of the "size" of the set of discontinuities of real functions.Īlso, " monsters" ( nowhere continuous functions, continuous but nowhere differentiable functions, space-filling curves) began to be investigated. Dedekind then constructed the real numbers by Dedekind cuts, in which irrational numbers are formally defined, which serve to fill the "gaps" between rational numbers, thereby creating a complete set: the continuum of real numbers, which had already been developed by Simon Stevin in terms of decimal expansions. ![]() Then, mathematicians started worrying that they were assuming the existence of a continuum of real numbers without proof. The last third of the century saw the arithmetization of analysis by Weierstrass, who thought that geometric reasoning was inherently misleading, and introduced the "epsilon-delta" definition of limit. In the middle of the 19th century Riemann introduced his theory of integration. The contributions of these mathematicians and others, such as Weierstrass, developed the (ε, δ)-definition of limit approach, thus founding the modern field of mathematical analysis. Poisson, Liouville, Fourier and others studied partial differential equations and harmonic analysis. He also introduced the concept of the Cauchy sequence, and started the formal theory of complex analysis. Thus, his definition of continuity required an infinitesimal change in x to correspond to an infinitesimal change in y. Instead, Cauchy formulated calculus in terms of geometric ideas and infinitesimals. In 1821, Cauchy began to put calculus on a firm logical foundation by rejecting the principle of the generality of algebra widely used in earlier work, particularly by Euler. Real analysis began to emerge as an independent subject when Bernard Bolzano introduced the modern definition of continuity in 1816, but Bolzano's work did not become widely known until the 1870s. In the 18th century, Euler introduced the notion of mathematical function. During this period, calculus techniques were applied to approximate discrete problems by continuous ones. Descartes and Fermat independently developed analytic geometry, and a few decades later Newton and Leibniz independently developed infinitesimal calculus, which grew, with the stimulus of applied work that continued through the 18th century, into analysis topics such as the calculus of variations, ordinary and partial differential equations, Fourier analysis, and generating functions. The modern foundations of mathematical analysis were established in 17th century Europe. His followers at the Kerala School of Astronomy and Mathematics further expanded his works, up to the 16th century. Alongside his development of the Taylor series of the trigonometric functions, he also estimated the magnitude of the error terms created by truncating these series and gave a rational approximation of an infinite series. In the 14th century, Madhava of Sangamagrama developed infinite series expansions, like the power series and the Taylor series, of functions such as sine, cosine, tangent and arctangent. The Indian mathematician Bhāskara II gave examples of the derivative and used what is now known as Rolle's theorem in the 12th century. Zu Chongzhi established a method that would later be called Cavalieri's principle to find the volume of a sphere in the 5th century. In Asia, the Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century AD to find the area of a circle. The explicit use of infinitesimals appears in Archimedes' The Method of Mechanical Theorems, a work rediscovered in the 20th century. Later, Greek mathematicians such as Eudoxus and Archimedes made more explicit, but informal, use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids. For instance, an infinite geometric sum is implicit in Zeno's paradox of the dichotomy. Early results in analysis were implicitly present in the early days of ancient Greek mathematics. Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. This was an early but informal example of a limit, one of the most basic concepts in mathematical analysis. Archimedes used the method of exhaustion to compute the area inside a circle by finding the area of regular polygons with more and more sides.
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