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Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times

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Compact space
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In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed and bounded. Examples include a closed interval, a rectangle, or a set of points. This notion is defined for general topological spaces than Euclidean space in various ways. One such generalization is that a space is compact if any infinite sequence of points sampled from the space must frequently get arbitrarily close to some point of the space. An equivalent definition is that every sequence of points must have an infinite subsequence that converges to some point of the space, the Heine-Borel theorem states that a subset of Euclidean space is compact in this sequential sense if and only if it is closed and bounded. Thus, if one chooses a number of points in the closed unit interval some of those points must get arbitrarily close to some real number in that space. For instance, some of the numbers 1/2, 4/5, 1/3, 5/6, 1/4, 6/7, the same set of points would not accumulate to any point of the open unit interval, so the open unit interval is not compact. Euclidean space itself is not compact since it is not bounded, in particular, the sequence of points 0, 1, 2, 3, … has no subsequence that converges to any given real number. Apart from closed and bounded subsets of Euclidean space, typical examples of compact spaces include spaces consisting not of geometrical points, the term compact was introduced into mathematics by Maurice Fréchet in 1904 as a distillation of this concept. Various equivalent notions of compactness, including sequential compactness and limit point compactness, in general topological spaces, however, different notions of compactness are not necessarily equivalent. This more subtle notion, introduced by Pavel Alexandrov and Pavel Urysohn in 1929, the term compact set is sometimes a synonym for compact space, but usually refers to a compact subspace of a topological space. In the 19th century, several disparate mathematical properties were understood that would later be seen as consequences of compactness. On the one hand, Bernard Bolzano had been aware that any bounded sequence of points has a subsequence that must eventually get close to some other point. The process could then be repeated by dividing the resulting smaller interval into smaller and smaller parts until it closes down on the limit point. The full significance of Bolzanos theorem, and its method of proof, in the 1880s, it became clear that results similar to the Bolzano–Weierstrass theorem could be formulated for spaces of functions rather than just numbers or geometrical points. The idea of regarding functions as points of a generalized space dates back to the investigations of Giulio Ascoli. The uniform limit of this sequence then played precisely the same role as Bolzanos limit point and this ultimately led to the notion of a compact operator as an offshoot of the general notion of a compact space. It was Maurice Fréchet who, in 1906, had distilled the essence of the Bolzano–Weierstrass property, in 1870, Eduard Heine showed that a continuous function defined on a closed and bounded interval was in fact uniformly continuous

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Banach space
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In mathematics, more specifically in functional analysis, a Banach space is a complete normed vector space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn, Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces, the vector space structure allows one to relate the behavior of Cauchy sequences to that of converging series of vectors. All norms on a vector space are equivalent. Every finite-dimensional normed space over R or C is a Banach space, if X and Y are normed spaces over the same ground field K, the set of all continuous K-linear maps T, X → Y is denoted by B. In infinite-dimensional spaces, not all maps are continuous. For Y a Banach space, the space B is a Banach space with respect to this norm, if X is a Banach space, the space B = B forms a unital Banach algebra, the multiplication operation is given by the composition of linear maps. If X and Y are normed spaces, they are isomorphic normed spaces if there exists a linear bijection T, X → Y such that T, if one of the two spaces X or Y is complete then so is the other space. Two normed spaces X and Y are isometrically isomorphic if in addition, T is an isometry, the Banach–Mazur distance d between two isomorphic but not isometric spaces X and Y gives a measure of how much the two spaces X and Y differ. Every normed space X can be embedded in a Banach space. More precisely, there is a Banach space Y and an isometric mapping T, X → Y such that T is dense in Y. If Z is another Banach space such that there is an isomorphism from X onto a dense subset of Z. This Banach space Y is the completion of the normed space X, the underlying metric space for Y is the same as the metric completion of X, with the vector space operations extended from X to Y. The completion of X is often denoted by X ^, the cartesian product X × Y of two normed spaces is not canonically equipped with a norm. However, several equivalent norms are used, such as ∥ ∥1 = ∥ x ∥ + ∥ y ∥, ∥ ∥ ∞ = max. In this sense, the product X × Y is complete if and only if the two factors are complete. If M is a linear subspace of a normed space X, there is a natural norm on the quotient space X / M

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Eduard Heine
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Heinrich Eduard Heine was a German mathematician. Heine became known for results on special functions and in real analysis, in particular, he authored an important treatise on spherical harmonics and Legendre functions. He also investigated basic hypergeometric series, heinrich Eduard Heine was born on 16 March 1821 in Berlin, as the eighth child of banker Karl Heine and his wife Henriette Märtens. Eduard was initially home schooled, then studied at the Friedrichswerdersche Gymnasium, in 1840 Heine returned to Berlin, where he studied mathematics under Peter Gustav Lejeune Dirichlet, while also attending classes of Jakob Steiner and Johann Franz Encke. In 1842 he was an awarded a Ph. D. by the University of Berlin for a thesis on differential equations submitted with Enno Dirksen, Heine dedicated the doctoral thesis to his professor Gustav Dirichlet. Next he went to the University of Königsberg to participate in the seminar of Carl Gustav Jacobi. In Königsberg Heine got in contact with fellow students Gustav Kirchhoff, in 1844 Heine went for a teaching position at the University of Bonn, passing his habilitation and starting as a privatdozent. He continued his research in mathematics in Bonn and, in 1848, was promoted to extraordinary professor, in 1850 he married Sophie Wolff, the daughter of a Berlin merchant, the couple had five children, four daughters and one son. In 1856 Heine moved as a professor to the University of Halle. From 1864 to 1865, he served as a rector of the university, in 1875 the University of Göttingen offered Heine a mathematics chair but he decided to reject the offer and remain in Halle. In 1877, at the centenary of Gauss birth, he was awarded the Gauss medal for his research, Eduard Heine died on 21 October 1881 in Halle. Robertson, Edmund F. Eduard Heine, MacTutor History of Mathematics archive, Eduard Heine at the Mathematics Genealogy Project