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Friedrich Riesz
Frigyes Riesz ( hu, Riesz Frigyes, , sometimes spelled as Frederic; 22 January 1880 – 28 February 1956) was a HungarianEberhard Zeidler: Nonlinear Functional Analysis and Its Applications: Linear monotone operators. Springer, 199/ref> mathematician who made fundamental contributions to functional analysis, as did his younger brother Marcel Riesz. Life and career He was born into a Jewish family in Győr, Austria-Hungary and died in Budapest, Hungary. Between 1911 and 1919 he was a professor at the Franz Joseph University in Kolozsvár, Austria-Hungary. The post-WW1 Treaty of Trianon transferred former Austro-Hungarian territory including Kolozsvár to the Kingdom of Romania, whereupon Kolozsvár's name changed to Cluj and the University of Kolozsvár moved to Szeged, Hungary, becoming the University of Szeged. Then, Riesz was the rector and a professor at the University of Szeged, as well as a member of the Hungarian Academy of Sciences. and the Polish Academy of Learning ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Győr
Győr ( , ; german: Raab, links=no; names of European cities in different languages: E-H#G, names in other languages) is the main city of northwest Hungary, the capital of Győr-Moson-Sopron County and Western Transdanubia, Western Transdanubia region, and – halfway between Budapest and Vienna – situated on one of the important roads of Central Europe. It is the sixth largest city in Hungary, and one of its seven main regional centres. The city has City with county rights, county rights. History The area along the Danube River has been inhabited by varying cultures since ancient times. The first large settlement dates back to the 5th century BCE; the inhabitants were Celts. They called the town ''Ara Bona'' "Good altar", later contracted to ''Arrabona'', a name which was used until the eighth century. Its shortened form is still used as the German (''Raab'') and Slovak (''Ráb'') names of the city. Roman merchants moved to Arrabona during the 1st century BCE. Around 10 CE, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lp Space
In mathematics, the spaces are function spaces defined using a natural generalization of the Norm (mathematics)#p-norm, -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Nicolas Bourbaki, Bourbaki group they were first introduced by Frigyes Riesz . spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Because of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used also in the theoretical discussion of problems in physics, statistics, economics, finance, engineering, and other disciplines. Applications Statistics In statistics, measures of central tendency and statistical dispersion, such as the mean, median, and standard deviation, are defined in terms of metrics, and measures of central tendency can be characterized as Central tendency#Solutions to variational problems, solutions to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History One of the earliest known mathematicians were Thales of Miletus (c. 624–c.546 BC); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. The number of known mathematicians grew when Pythagoras of Samos (c. 582–c. 507 BC) established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman mathematician recorded by history was Hypati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hungarian People
Hungarians, also known as Magyars ( ; hu, magyarok ), are a nation and ethnic group native to Hungary () and Kingdom of Hungary, historical Hungarian lands who share a common Hungarian culture, culture, Hungarian history, history, Magyar tribes, ancestry, and Hungarian language, language. The Hungarian language belongs to the Uralic languages, Uralic language family. There are an estimated 15 million ethnic Hungarians and their descendants worldwide, of whom 9.6 million live in today's Hungary. About 2–3 million Hungarians live in areas that were part of the Kingdom of Hungary before the Treaty of Trianon in 1920 and are now parts of Hungary's seven neighbouring countries, Hungarians in Slovakia, Slovakia, Hungarians in Ukraine, Ukraine, Hungarians in Romania, Romania, Hungarians in Serbia, Serbia, Hungarians of Croatia, Croatia, Prekmurje, Slovenia, and Hungarians in Austria, Austria. Hungarian diaspora, Significant groups of people with Hungarian ancestry live in various oth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riesz–Markov–Kakutani Representation Theorem
In mathematics, the Riesz–Markov–Kakutani representation theorem relates linear functionals on spaces of continuous functions on a locally compact space to measures in measure theory. The theorem is named for who introduced it for continuous functions on the unit interval, who extended the result to some non-compact spaces, and who extended the result to compact Hausdorff spaces. There are many closely related variations of the theorem, as the linear functionals can be complex, real, or positive, the space they are defined on may be the unit interval or a compact space or a locally compact space, the continuous functions may be vanishing at infinity or have compact support, and the measures can be Baire measures or regular Borel measures or Radon measures or signed measures or complex measures. The representation theorem for positive linear functionals on ''Cc''(''X'') The following theorem represents positive linear functionals on ''Cc''(''X''), the space of conti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radon–Riesz Property
The Radon–Riesz property is a mathematical property for normed spaces that helps ensure convergence in norm. Given two assumptions (essentially weak convergence and continuity of norm), we would like to ensure convergence in the norm topology. Definition Suppose that (''X'', , , ·, , ) is a normed space. We say that ''X'' has the ''Radon–Riesz property'' (or that ''X'' is a ''Radon–Riesz space'') if whenever (x_) is a sequence in the space and x is a member of ''X'' such that (x_) converges weakly to x and \lim_ \Vert x_n \Vert = \Vert x\Vert , then (x_) converges to x in norm; that is, \lim_ \Vert x_n - x\Vert = 0 . Other names Although it would appear that Johann Radon was one of the first to make significant use of this property in 1913, M. I. Kadets and V. L. Klee also used versions of the Radon–Riesz property to make advancements in Banach space theory in the late 1920s. It is common for the Radon–Riesz property to also be referred to as the Kadets–Klee propert ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riesz Projector
In mathematics, or more specifically in spectral theory, the Riesz projector is the projector onto the eigenspace corresponding to a particular eigenvalue of an operator (or, more generally, a projector onto an invariant subspace corresponding to an isolated part of the spectrum). It was introduced by Frigyes Riesz in 1912. Definition Let A be a closed linear operator in the Banach space \mathfrak. Let \Gamma be a simple or composite rectifiable contour, which encloses some region G_\Gamma and lies entirely within the resolvent set \rho(A) (\Gamma\subset\rho(A)) of the operator A. Assuming that the contour \Gamma has a positive orientation with respect to the region G_\Gamma, the Riesz projector corresponding to \Gamma is defined by : P_\Gamma=-\frac\oint_\Gamma(A-z I_)^\,\mathrmz; here I_ is the identity operator in \mathfrak. If \lambda\in\sigma(A) is the only point of the spectrum of A in G_\Gamma, then P_\Gamma is denoted by P_\lambda. Properties The operator P_\Gamma is a p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riesz–Fischer Theorem
In mathematics, the Riesz–Fischer theorem in real analysis is any of a number of closely related results concerning the properties of the space ''L''2 of square integrable functions. The theorem was proven independently in 1907 by Frigyes Riesz and Ernst Sigismund Fischer. For many authors, the Riesz–Fischer theorem refers to the fact that the Lp spaces L^p from Lebesgue integration theory are complete. Modern forms of the theorem The most common form of the theorem states that a measurable function on \pi, \pi/math> is square integrable if and only if the corresponding Fourier series converges in the Lp space L^2. This means that if the ''N''th partial sum of the Fourier series corresponding to a square-integrable function ''f'' is given by S_N f(x) = \sum_^ F_n \, \mathrm^, where F_n, the ''n''th Fourier coefficient, is given by F_n =\frac\int_^\pi f(x)\, \mathrm^\, \mathrmx, then \lim_ \left\Vert S_N f - f \right\, _2 = 0, where \, \,\cdot\,\, _2 is the L^2-norm. Conve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riesz Representation Theorem
:''This article describes a theorem concerning the dual of a Hilbert space. For the theorems relating linear functionals to measures, see Riesz–Markov–Kakutani representation theorem.'' The Riesz representation theorem, sometimes called the Riesz–Fréchet representation theorem after Frigyes Riesz and Maurice René Fréchet, establishes an important connection between a Hilbert space and its continuous dual space. If the underlying field is the real numbers, the two are isometrically isomorphic; if the underlying field is the complex numbers, the two are isometrically anti-isomorphic. The (anti-) isomorphism is a particular natural isomorphism. Preliminaries and notation Let H be a Hilbert space over a field \mathbb, where \mathbb is either the real numbers \R or the complex numbers \Complex. If \mathbb = \Complex (resp. if \mathbb = \R) then H is called a (resp. a ). Every real Hilbert space can be extended to be a dense subset of a unique (up to bijective isometry) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riesz's Lemma
Riesz's lemma (after Frigyes Riesz) is a lemma in functional analysis. It specifies (often easy to check) conditions that guarantee that a subspace in a normed vector space is dense. The lemma may also be called the Riesz lemma or Riesz inequality. It can be seen as a substitute for orthogonality when one is not in an inner product space. The result Riesz's Lemma. Let ''X'' be a normed space, ''Y'' be a closed proper subspace of ''X'' and α be a real number with Then there exists an ''x'' in ''X'' with , ''x'', = 1 such that , ''x'' − ''y'', ≥ α for all ''y'' in ''Y''. ''Remark 1.'' For the finite-dimensional case, equality can be achieved. In other words, there exists ''x'' of unit norm such that ''d''(''x'', ''Y'') = 1. When dimension of ''X'' is finite, the unit ball ''B'' ⊂ ''X'' is compact. Also, the distance function ''d''(· , ''Y'') is continuous. Therefore its image on the unit ball ''B'' must be a compac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riesz Rearrangement Inequality
In mathematics, the Riesz rearrangement inequality, sometimes called Riesz–Sobolev inequality, states that any three non-negative functions f : \mathbb^n \to \mathbb^+, g : \mathbb^n \to \mathbb^+ and h : \mathbb^n \to \mathbb^+ satisfy the inequality :\iint_ f(x) g(x-y) h(y) \, dx\,dy \le \iint_ f^*(x) g^*(x-y) h^*(y) \, dx\,dy, where f^* : \mathbb^n \to \mathbb^+, g^* : \mathbb^n \to \mathbb^+ and h^* : \mathbb^n \to \mathbb^+ are the symmetric decreasing rearrangements of the functions f, g and h respectively. History The inequality was first proved by Frigyes Riesz in 1930, and independently reproved by S.L.Sobolev in 1938. Brascamp, Lieb and Luttinger have shown that can be generalized to arbitrarily (but finitely) many functions acting on arbitrarily many variables. Applications The Riesz rearrangement inequality can be used to prove the Pólya–Szegő inequality. Proofs One-dimensional case In the one-dimensional case, the inequality is first proved when th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riesz Space
In mathematics, a Riesz space, lattice-ordered vector space or vector lattice is a partially ordered vector space where the order structure is a lattice. Riesz spaces are named after Frigyes Riesz who first defined them in his 1928 paper ''Sur la décomposition des opérations fonctionelles linéaires''. Riesz spaces have wide-ranging applications. They are important in measure theory, in that important results are special cases of results for Riesz spaces. For example, the Radon–Nikodym theorem follows as a special case of the Freudenthal spectral theorem. Riesz spaces have also seen application in mathematical economics through the work of Greek-American economist and mathematician Charalambos D. Aliprantis. Definition Preliminaries If X is an ordered vector space (which by definition is a vector space over the reals) and if S is a subset of X then an element b \in X is an upper bound (resp. lower bound) of S if s \leq b (resp. s \geq b) for all s \in S. An element ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
