Stefan Banach
Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a Polish mathematician who is generally considered one of the 20th century's most important and influential mathematicians. He was the founder of modern functional analysis, and an original member of the Lwów School of Mathematics. His major work was the 1932 book, ''Théorie des opérations linéaires'' (Theory of Linear Operations), the first monograph on the general theory of functional analysis. Born in Kraków to a family of Goral descent, Banach showed a keen interest in mathematics and engaged in solving mathematical problems during school recess. After completing his secondary education, he befriended Hugo Steinhaus, with whom he established the Polish Mathematical Society in 1919 and later published the scientific journal '' Studia Mathematica''. In 1920, he received an assistantship at the Lwów Polytechnic, subsequently becoming a professor in 1922 and a member of the Polish Academy of Learning in 1924. Banach ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Kraków
Kraków (), or Cracow, is the second-largest and one of the oldest cities in Poland. Situated on the Vistula River in Lesser Poland Voivodeship, the city dates back to the seventh century. Kraków was the official capital of Poland until 1596 and has traditionally been one of the leading centres of Polish academic, economic, cultural and artistic life. Cited as one of Europe's most beautiful cities, its Old Town with Wawel Royal Castle was declared a UNESCO World Heritage Site in 1978, one of the first 12 sites granted the status. The city has grown from a Stone Age settlement to Poland's second-most-important city. It began as a hamlet on Wawel Hill and was reported by Ibrahim Ibn Yakoub, a merchant from Cordoba, as a busy trading centre of Central Europe in 985. With the establishment of new universities and cultural venues at the emergence of the Second Polish Republic in 1918 and throughout the 20th century, Kraków reaffirmed its role as a major national academic and a ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Banach–Tarski Paradox
The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. Indeed, the reassembly process involves only moving the pieces around and rotating them without changing their shape. However, the pieces themselves are not "solids" in the usual sense, but infinite scatterings of points. The reconstruction can work with as few as five pieces. An alternative form of the theorem states that given any two "reasonable" solid objects (such as a small ball and a huge ball), the cut pieces of either one can be reassembled into the other. This is often stated informally as "a pea can be chopped up and reassembled into the Sun" and called the "pea and the Sun paradox". The theorem is called a paradox because it contradicts ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Gorals
The Gorals ( pl, Górale; Goral dialect: ''Górole''; sk, Gorali; Cieszyn Silesia dialect, Cieszyn Silesian: ''Gorole''), also known as the Highlanders (in Poland as the Polish Highlanders) are an indigenous ethnographic or ethnic group primarily found in their traditional area of southern Poland, northern Slovakia and in the region of Cieszyn Silesia in the Czech Republic, where they are known as the Silesian Gorals. There is also a significant Goral diaspora in the area of Bukovina in western Ukraine and in northern Romania, as well as in Chicago, the seat of the Polish Highlanders Alliance of North America. History In the 13th century, Vlach shepherds migrated to the Divisions of the Carpathians#Western Carpathians (province), Western Carpathian mountains, gradually moving northwest from the Balkans and settling on History of Poland during the Piast dynasty, Polish lands there. In the 16th and 17th centuries, Gorals settled the upper Kysuca and Orava (river), Orava rivers an ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Lwów School Of Mathematics
The Lwów school of mathematics ( pl, lwowska szkoła matematyczna) was a group of Polish mathematicians who worked in the interwar period in Lwów, Poland (since 1945 Lviv, Ukraine). The mathematicians often met at the famous Scottish Café to discuss mathematical problems, and published in the journal '' Studia Mathematica'', founded in 1929. The school was renowned for its productivity and its extensive contributions to subjects such as point-set topology, set theory and functional analysis. The biographies and contributions of these mathematicians were documented in 1980 by their contemporary Kazimierz Kuratowski in his book ''A Half Century of Polish Mathematics: Remembrances and Reflections''. Members Notable members of the Lwów school of mathematics included: * Stefan Banach * Feliks Barański * Władysław Orlicz * Stanisław Saks * Hugo Steinhaus * Stanisław Mazur * Stanisław Ulam * Józef Schreier * Juliusz Schauder * Mark Kac * Antoni Łomnicki * Stefan Kaczmar ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Functional Analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, Topological space#Definition, topology, etc.) and the linear transformation, linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of function space, spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous function, continuous, unitary operator, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential equations, differential and integral equations. The usage of the word ''functional (mathematics), functional'' as a noun goes back to the calculus of variati ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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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]   |
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Poland
Poland, officially the Republic of Poland, is a country in Central Europe. It is divided into 16 administrative provinces called voivodeships, covering an area of . Poland has a population of over 38 million and is the fifth-most populous member state of the European Union. Warsaw is the nation's capital and largest metropolis. Other major cities include Kraków, Wrocław, Łódź, Poznań, Gdańsk, and Szczecin. Poland has a temperate transitional climate and its territory traverses the Central European Plain, extending from Baltic Sea in the north to Sudeten and Carpathian Mountains in the south. The longest Polish river is the Vistula, and Poland's highest point is Mount Rysy, situated in the Tatra mountain range of the Carpathians. The country is bordered by Lithuania and Russia to the northeast, Belarus and Ukraine to the east, Slovakia and the Czech Republic to the south, and Germany to the west. It also shares maritime boundaries with Denmark and Sweden. ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Polish Academy Of Learning
The Polish Academy of Arts and Sciences or Polish Academy of Learning ( pl, Polska Akademia Umiejętności), headquartered in Kraków and founded in 1872, is one of two institutions in contemporary Poland having the nature of an academy of sciences. (The other is the Polish Academy of Sciences, headquartered in Warsaw.) The Polish Academy of Arts and Sciences is co-owner of the Polish Library in Paris. History The Academy traces its origins to Academy of Learning founded in 1871, itself a result of the transformation of the , in existence since 1815. Though formally limited to the Austrian Partition, the Academy served from the beginning as a learned and cultural society for the entire Polish nation. Its activities extended beyond the boundaries of the Austrian Partition, gathering scholars from all of Poland, and many other countries as well. Some indication of how the Academy's influence extended beyond the boundaries of the Partitions came in 1893, when the collection of the ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Surjection Of Fréchet Spaces
The theorem on the surjection of Fréchet spaces is an important theorem, due to Stefan Banach, that characterizes when a continuous linear operator between Fréchet spaces is surjective. The importance of this theorem is related to the open mapping theorem, which states that a continuous linear surjection between Fréchet spaces is an open map. Often in practice, one knows that they have a continuous linear map between Fréchet spaces and wishes to show that it is surjective in order to use the open mapping theorem to deduce that it is also an open mapping. This theorem may help reach that goal. Preliminaries, definitions, and notation Let L : X \to Y be a continuous linear map between topological vector spaces. The continuous dual space of X is denoted by X^. The transpose of L is the map ^t L : Y^ \to X^ defined by L \left(y^\right) := y^ \circ L. If L : X \to Y is surjective then ^t L : Y^ \to X^ will be injective, but the converse is not true in general. The weak to ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Banach–Alaoglu Theorem
In functional analysis and related branches of mathematics, the Banach–Alaoglu theorem (also known as Alaoglu's theorem) states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. A common proof identifies the unit ball with the weak-* topology as a closed subset of a product of compact sets with the product topology. As a consequence of Tychonoff's theorem, this product, and hence the unit ball within, is compact. This theorem has applications in physics when one describes the set of states of an algebra of observables, namely that any state can be written as a convex linear combination of so-called pure states. History According to Lawrence Narici and Edward Beckenstein, the Alaoglu theorem is a "very important result - maybe most important fact about the weak-* topology - hatechos throughout functional analysis." In 1912, Helly proved that the unit ball of the continuous dual space of C( , b is countably weak-* compact. ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Hahn–Banach Theorem
The Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed vector space to make the study of the dual space "interesting". Another version of the Hahn–Banach theorem is known as the Hahn–Banach separation theorem or the hyperplane separation theorem, and has numerous uses in convex geometry. History The theorem is named for the mathematicians Hans Hahn and Stefan Banach, who proved it independently in the late 1920s. The special case of the theorem for the space C[a, b] of continuous functions on an interval was proved earlier (in 1912) by Eduard Helly, and a more general extension theorem, the M. Riesz extension theorem, from which the Hahn–Banach theorem can be derived, was proved in 1923 by Marcel Riesz. The first Hahn–Banach theorem was proved by ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Banach–Schauder Theorem
In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem or the Banach theorem (named after Stefan Banach and Juliusz Schauder), is a fundamental result which states that if a bounded or continuous linear operator between Banach spaces is surjective then it is an open map. Classical (Banach space) form This proof uses the Baire category theorem, and completeness of both X and Y is essential to the theorem. The statement of the theorem is no longer true if either space is just assumed to be a normed space, but is true if X and Y are taken to be Fréchet spaces. Suppose A : X \to Y is a surjective continuous linear operator. In order to prove that A is an open map, it is sufficient to show that A maps the open unit ball in X to a neighborhood of the origin of Y. Let U = B_1^X(0), V = B_1^Y(0). Then X = \bigcup_ k U. Since A is surjective: Y = A(X) = A\left(\bigcup_ k U\right) = \bigcup_ A(kU). But Y is Banach so by Baire's category th ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |