1.
Brouwer fixed-point theorem
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Brouwers fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Brouwer. It states that for any function f mapping a compact convex set into itself there is a point x 0 such that f = x 0. The simplest forms of Brouwers theorem are for continuous functions f from a closed interval I in the numbers to itself or from a closed disk D to itself. A more general form than the latter is for continuous functions from a compact subset K of Euclidean space to itself. Among hundreds of fixed-point theorems, Brouwers is particularly well known, in its original field, this result is one of the key theorems characterizing the topology of Euclidean spaces, along with the Jordan curve theorem, the hairy ball theorem and the Borsuk–Ulam theorem. This gives it a place among the theorems of topology. The theorem is used for proving deep results about differential equations and is covered in most introductory courses on differential geometry. It appears in unlikely fields such as game theory, the theorem was first studied in view of work on differential equations by the French mathematicians around Poincaré and Picard. Proving results such as the Poincaré–Bendixson theorem requires the use of topological methods and this work at the end of the 19th century opened into several successive versions of the theorem. The general case was first proved in 1910 by Jacques Hadamard, the theorem has several formulations, depending on the context in which it is used and its degree of generalization. The simplest is sometimes given as follows, In the plane Every continuous function from a disk to itself has at least one fixed point. This can be generalized to an arbitrary dimension, In Euclidean space Every continuous function from a closed ball of a Euclidean space into itself has a fixed point. A slightly more general version is as follows, Convex compact set Every continuous function from a compact subset K of a Euclidean space to K itself has a fixed point. The theorem holds only for sets that are compact and convex, the following examples show why these requirements are important. Consider the function f = x +1 which is a function from R to itself. As it shifts every point to the right, it cannot have a fixed point, note that R is convex and closed, but not bounded. Consider the function f = x +12 which is a function from the open interval to itself. In this interval, it shifts every point to the right, note that is convex and bounded, but not closed

2.
Andrey Nikolayevich Tikhonov
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Andrey Nikolayevich Tikhonov was a Soviet and Russian mathematician and geophysicist known for important contributions to topology, functional analysis, mathematical physics, and ill-posed problems. He was also one of the inventors of the method in geophysics. Tikhonovs surname is transliterated as Tychonoff, because he originally published in German. Other forms of his name include Tychonov, Tihonov, Tichonov, born near Smolensk, he studied at the Moscow State University where he received Ph. D. in 1927 under direction of Pavel Sergeevich Alexandrov. In 1933 he was appointed as a professor at Moscow State University and he became a corresponding member of the USSR Academy of Sciences on 29 January 1939 and a full member of the USSR Academy of Sciences on 1 July 1966. Tikhonov worked in a number of different fields in mathematics and he made important contributions to topology, functional analysis, mathematical physics, and certain classes of ill-posed problems. Tikhonov regularization, one of the most widely used methods to solve ill-posed inverse problems, is named in his honor, in his honor, completely regular topological spaces are also named Tychonoff spaces. In mathematical physics, he proved the fundamental uniqueness theorems for the heat equation, in asymptotical analysis, he founded the theory of asymptotic analysis for differential equations with small parameter in the leading derivative. Tikhonov played the role in founding the Faculty of Computational Mathematics and Cybernetics of Moscow State University. Tikhonov received numerous honors and awards for his work, including the Lenin Prize, Tikhonov, The Theory of Functions of a Complex Variable, Mir Publishers, English translation,1978. Arsenin, Solutions of Ill-Posed Problems, Winston, New York,1977, goncharsky, Ill-posed Problems in the Natural Sciences, Oxford University Press, Oxford,1987. Samarskii, Equations of Mathematical Physics, Dover Publications,1990, stepanov, A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems, Kluwer, Dordrecht,1995. Nonlinear Ill-Posed Problems, Chapman and Hall, London, Weinheim, New York, Tokyo, Melbourne, Madras, oConnor, John J. Robertson, Edmund F. Andrey Nikolayevich Tikhonov, MacTutor History of Mathematics archive, University of St Andrews. Andrey Nikolayevich Tikhonov at the Mathematics Genealogy Project

3.
Encyclopedia of Mathematics
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The Encyclopedia of Mathematics is a large reference work in mathematics. It is available in form and on CD-ROM. The 2002 version contains more than 8,000 entries covering most areas of mathematics at a level. The encyclopedia is edited by Michiel Hazewinkel and was published by Kluwer Academic Publishers until 2003, the CD-ROM contains animations and three-dimensional objects. Until November 29,2011, a version of the encyclopedia could be browsed online free of charge online This URL now redirects to the new wiki incarnation of the EOM. A new dynamic version of the encyclopedia is now available as a public wiki online and this new wiki is a collaboration between Springer and the European Mathematical Society. This new version of the encyclopedia includes the entire contents of the online version. All entries will be monitored for content accuracy by members of a board selected by the European Mathematical Society. Vinogradov, I. M. Matematicheskaya entsiklopediya, Moscow, Sov, Hazewinkel, M. Encyclopaedia of Mathematics, Kluwer,1994. Hazewinkel, M. Encyclopaedia of Mathematics, Vol.1, Hazewinkel, M. Encyclopaedia of Mathematics, Vol.2, Kluwer,1988. Hazewinkel, M. Encyclopaedia of Mathematics, Vol.3, Hazewinkel, M. Encyclopaedia of Mathematics, Vol.4, Kluwer,1989. Hazewinkel, M. Encyclopaedia of Mathematics, Vol.5, Hazewinkel, M. Encyclopaedia of Mathematics, Vol.6, Kluwer,1990. Hazewinkel, M. Encyclopaedia of Mathematics, Vol.7, Hazewinkel, M. Encyclopaedia of Mathematics, Vol.8, Kluwer,1992. Hazewinkel, M. Encyclopaedia of Mathematics, Vol.9, Hazewinkel, M. Encyclopaedia of Mathematics, Vol.10, Kluwer,1994. Hazewinkel, M. Encyclopaedia of Mathematics, Supplement I, Kluwer,1997, Hazewinkel, M. Encyclopaedia of Mathematics, Supplement II, Kluwer,2000. Hazewinkel, M. Encyclopaedia of Mathematics, Supplement III, Kluwer,2002, Hazewinkel, M. Encyclopaedia of Mathematics on CD-ROM, Kluwer,1998. Encyclopedia of Mathematics, public wiki monitored by a board under the management of the European Mathematical Society. List of online encyclopedias Current page of M. Hazewinkel Online Encyclopedia of Mathematics

4.
Hilbert space
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The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of algebra and calculus from the two-dimensional Euclidean plane. A Hilbert space is a vector space possessing the structure of an inner product that allows length. Furthermore, Hilbert spaces are complete, there are limits in the space to allow the techniques of calculus to be used. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces, the earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis —and ergodic theory, john von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era for functional analysis, geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space, at a deeper level, perpendicular projection onto a subspace plays a significant role in optimization problems and other aspects of the theory. An element of a Hilbert space can be specified by its coordinates with respect to a set of coordinate axes. When that set of axes is countably infinite, this means that the Hilbert space can also usefully be thought of in terms of the space of sequences that are square-summable. The latter space is often in the literature referred to as the Hilbert space. One of the most familiar examples of a Hilbert space is the Euclidean space consisting of vectors, denoted by ℝ3. The dot product takes two vectors x and y, and produces a real number x·y, If x and y are represented in Cartesian coordinates, then the dot product is defined by ⋅ = x 1 y 1 + x 2 y 2 + x 3 y 3. The dot product satisfies the properties, It is symmetric in x and y, x · y = y · x. It is linear in its first argument, · y = ax1 · y + bx2 · y for any scalars a, b, and vectors x1, x2, and y. It is positive definite, for all x, x · x ≥0, with equality if. An operation on pairs of vectors that, like the dot product, a vector space equipped with such an inner product is known as a inner product space. Every finite-dimensional inner product space is also a Hilbert space, multivariable calculus in Euclidean space relies on the ability to compute limits, and to have useful criteria for concluding that limits exist

5.
Jean Leray
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Jean Leray was a French mathematician, who worked on both partial differential equations and algebraic topology. He studied at École Normale Supérieure from 1926 to 1929 and he received his Ph. D. in 1933. Leray wrote an important paper that founded the study of solutions of the Navier–Stokes equations. From 1938 to 1939 he was professor at the University of Nancy and he did not join the Bourbaki group, although he was close with its founders. His main work in topology was carried out while he was in a prisoner of war camp in Edelbach and he concealed his expertise on differential equations, fearing that its connections with applied mathematics could lead him to be asked to do war work. Lerays work of this period proved seminal to the development of spectral sequences and sheaves and these were subsequently developed by many others, each separately becoming an important tool in homological algebra. He returned to work on differential equations from about 1950. He was professor at the University of Paris from 1945 to 1947 and he was awarded the Malaxa Prize, the Grand Prix in mathematical sciences, the Feltrinelli Prize, the Wolf Prize in Mathematics, and the Lomonosov Gold Medal. Leray spectral sequence Leray cover Lerays theorem Leray–Hirsch theorem OConnor, John J. Robertson, Jean Leray, MacTutor History of Mathematics archive, University of St Andrews. Jean Leray at the Mathematics Genealogy Project Jean Leray, by Armand Borel, Gennadi M. Henkin, and Peter D. Lax, Notices of the American Mathematical Society, vol

6.
Neil Trudinger
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Neil Sidney Trudinger is an Australian mathematician, known particularly for his work in the field of nonlinear elliptic partial differential equations. After completing his B. Sc at the University of New England in 1962 and he was awarded a Ph. D in 1966 for his thesis Quasilinear Elliptical Partial Differential Equations in n Variables. He then returned to Australia where he was appointed as a lecturer at Macquarie University in 1967, in 1970, he moved to University of Queensland where he was first appointed as a Reader, then as Professor. In 1973 he moved to the Australian National University, in 2016 he moved to the University of Wollongong, where he is currently appointed as a Distinguished Professor. He currently coordinates ANUs Applied and Nonlinear Analysis program and he is co-author, together with his thesis advisor, David Gilbarg, of the book Elliptic Partial Differential Equations of Second Order. 1978, elected as a fellow of the Australian Academy of Science,1981, first recipient of the Australian Mathematical Society Medal. 1996, awarded the Hannan Medal of the Australian Academy of Science,1997, elected as a Fellow of the Royal Society of London. 2008, awarded the Leroy P. Steele Prize for Mathematical Exposition by the American Mathematical Society,2012, elected as a fellow of the American Mathematical Society. 2014, gave the Łojasiewicz Lecture at the Jagiellonian University in Kraków, trudingers theorem Yamabe problem MacTutor biography Neil Trudinger at the Mathematics Genealogy Project ANU Applied and Nonlinear Analysis program

7.
Michiel Hazewinkel
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Born in Amsterdam to Jan Hazewinkel and Geertrude Hendrika Werner, Hazewinkel studied at the University of Amsterdam. After graduation Hazewinkel started his career as Assistant Professor at the University of Amsterdam in 1969. In 1970 he became Associate Professor at the Erasmus University Rotterdam, here he was thesis advisor of Roelof Stroeker, M. van de Vel, Jo Ritzen, and Gerard van der Hoek. From 1973 to 1975 he was also Professor at the Universitaire Instelling Antwerpen, were Marcel van de Vel was his PhD student. At the Centre for Mathematics and Computer CWI in Amsterdam in 1988 he became Professor of Mathematics and head of the Department of Algebra, Analysis, in 1994 Hazewinkel was elected member of the International Academy of Computer Sciences and Systems. Hazewinkel has authored and edited books, and numerous articles. With Michel Demazure and Pierre Gabriel, on invariants, canonical forms and moduli for linear, constant, finite dimensional, dynamical systems. Moduli and canonical forms for linear dynamical systems II, The topological case, on Lie algebras and finite dimensional filtering. Stochastics, a journal of probability and stochastic processes 7. 1–2. Nonexistence of finite-dimensional filters for conditional statistics of the sensor problem. Systems & control letters 3.6, 331–340, the algebra of quasi-symmetric functions is free over the integers

8.
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

9.
Stereotype space
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In functional analysis and related areas of mathematics stereotype spaces are topological vector spaces defined by a special variant of reflexivity condition. Each pseudocomplete barreled space X is stereotype, a metrizable locally convex space X is stereotype if and only if X is complete. Each infinite dimensional normed space X considered with the X ⋆ -weak topology is not stereotype, there exist stereotype spaces which are not Mackey spaces. Some simple connections between the properties of a stereotype space X and those of its dual space X ⋆ are expressed in the following list of regularities, the first results on this type of reflexivity of topological vector spaces were obtained by M. F. Smith in 1952. Further investigations were conducted by B. S. Brudovskii, W. C, waterhouse, K. Brauner, S. S. Akbarov, and E. T. Shavgulidze. Each locally convex space X can be transformed into a space with the help of the standard operations of pseudocompletion and pseudosaturation defined by the following two propositions. If X is a locally convex space, then its pseudosaturation X △ is stereotype. Dually, if X is a locally convex space, then its pseudocompletion X ▽ is stereotype. For arbitrary locally convex space X the spaces X △ ▽ and X ▽ △ are stereotype and it defines two natural tensor products X ⊛ Y, = Hom ⋆, X ⊙ Y, = Hom. This condition is weaker than the existence of the Schauder basis, the following proposition holds, If two stereotype spaces X and Y have the stereotype approximation property, then the spaces Hom, X ⊛ Y and X ⊙ Y have the stereotype approximation property as well. In particular, if X has the approximation property, then the same is true for X ⋆. This allows to reduce the list of counterexamples in comparison with the Banach theory, the arising theory of stereotype algebras allows to simplify constructions in the duality theories for non-commutative groups. In particular, the group algebras in these theories become Hopf algebras in the algebraic sense. Schaefer, Helmuth H. Topological vector spaces, Robertson, A. P. Robertson, W. J. Topological vector spaces. The Pontrjagin duality theorem in linear spaces, on k- and c-reflexivity of locally convex vector spaces. Brauner, K. Duals of Fréchet spaces and a generalization of the Banach-Dieudonné theorem, Akbarov, S. S. Pontryagin duality in the theory of topological vector spaces and in topological algebra. Akbarov, S. S. Holomorphic functions of exponential type, envelopes and refinements in categories, with applications to functional analysis. On two classes of spaces reflexive in the sense of Pontryagin, Akbarov, S. S. Pontryagin duality and topological algebras

10.
Inner product space
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In linear algebra, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a quantity known as the inner product of the vectors. Inner products allow the introduction of intuitive geometrical notions such as the length of a vector or the angle between two vectors. They also provide the means of defining orthogonality between vectors, inner product spaces generalize Euclidean spaces to vector spaces of any dimension, and are studied in functional analysis. An inner product induces a associated norm, thus an inner product space is also a normed vector space. A complete space with a product is called a Hilbert space. An space with a product is called a pre-Hilbert space, since its completion with respect to the norm induced by the inner product is a Hilbert space. Inner product spaces over the field of numbers are sometimes referred to as unitary spaces. In this article, the field of scalars denoted F is either the field of real numbers R or the field of complex numbers C, formally, an inner product space is a vector space V over the field F together with an inner product, i. e. Some authors, especially in physics and matrix algebra, prefer to define the inner product, then the first argument becomes conjugate linear, rather than the second. In those disciplines we would write the product ⟨ x, y ⟩ as ⟨ y | x ⟩, respectively y † x. Here the kets and columns are identified with the vectors of V and this reverse order is now occasionally followed in the more abstract literature, taking ⟨ x, y ⟩ to be conjugate linear in x rather than y. A few instead find a ground by recognizing both ⟨ ⋅, ⋅ ⟩ and ⟨ ⋅ | ⋅ ⟩ as distinct notations differing only in which argument is conjugate linear. There are various reasons why it is necessary to restrict the basefield to R and C in the definition. Briefly, the basefield has to contain an ordered subfield in order for non-negativity to make sense, the basefield has to have additional structure, such as a distinguished automorphism. More generally any quadratically closed subfield of R or C will suffice for this purpose, however in these cases when it is a proper subfield even finite-dimensional inner product spaces will fail to be metrically complete. In contrast all finite-dimensional inner product spaces over R or C, such as used in quantum computation, are automatically metrically complete. In some cases we need to consider non-negative semi-definite sesquilinear forms and this means that ⟨ x, x ⟩ is only required to be non-negative

11.
Euclidean space
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In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces. It is named after the Ancient Greek mathematician Euclid of Alexandria, the term Euclidean distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions, classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. Geometric constructions are used to define rational numbers. It means that points of the space are specified with collections of real numbers and this approach brings the tools of algebra and calculus to bear on questions of geometry and has the advantage that it generalizes easily to Euclidean spaces of more than three dimensions. From the modern viewpoint, there is only one Euclidean space of each dimension. With Cartesian coordinates it is modelled by the coordinate space of the same dimension. In one dimension, this is the line, in two dimensions, it is the Cartesian plane, and in higher dimensions it is a coordinate space with three or more real number coordinates. One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance, for example, there are two fundamental operations on the plane. One is translation, which means a shifting of the plane so that point is shifted in the same direction. The other is rotation about a point in the plane. In order to all of this mathematically precise, the theory must clearly define the notions of distance, angle, translation. Even when used in theories, Euclidean space is an abstraction detached from actual physical locations, specific reference frames, measurement instruments. The standard way to such space, as carried out in the remainder of this article, is to define the Euclidean plane as a two-dimensional real vector space equipped with an inner product. The reason for working with vector spaces instead of Rn is that it is often preferable to work in a coordinate-free manner. Once the Euclidean plane has been described in language, it is actually a simple matter to extend its concept to arbitrary dimensions. For the most part, the vocabulary, formulae, and calculations are not made any more difficult by the presence of more dimensions. Intuitively, the distinction says merely that there is no choice of where the origin should go in the space