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Stanisław Mazur
Stanisław Mieczysław Mazur (1 January 1905, Lwów – 5 November 1981, Warsaw) was a Polish mathematician and a member of the Polish Academy of Sciences. Mazur made important contributions to geometrical methods in linear and nonlinear functional analysis and to the study of Banach algebras. He was also interested in summability theory, infinite games and computable functions. Lwów and Warsaw Mazur was a student of Stefan Banach at University of Lwów. His doctorate, under Banach's supervision, was awarded in 1935. Mazur, with Juliusz Schauder, was an Invited Speaker of the ICM in 1936 in Oslo. Mazur was a close collaborator with Banach at Lwów and was a member of the Lwów School of Mathematics, where he participated in the mathematical activities at the Scottish Café. On 6 November 1936, he posed the " basis problem" of determining whether every Banach space has a Schauder basis, with Mazur promising a "live goose" as a reward: 37 years later and in a ceremony that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lviv
Lviv ( uk, Львів) is the largest city in western Ukraine, and the seventh-largest in Ukraine, with a population of . It serves as the administrative centre of Lviv Oblast and Lviv Raion, and is one of the main cultural centres of Ukraine. It was named in honour of Leo, the eldest son of Daniel, King of Ruthenia. Lviv emerged as the centre of the historical regions of Red Ruthenia and Galicia in the 14th century, superseding Halych, Chełm, Belz and Przemyśl. It was the capital of the Kingdom of Galicia–Volhynia from 1272 to 1349, when it was conquered by King Casimir III the Great of Poland. From 1434, it was the regional capital of the Ruthenian Voivodeship in the Kingdom of Poland. In 1772, after the First Partition of Poland, the city became the capital of the Habsburg Kingdom of Galicia and Lodomeria. In 1918, for a short time, it was the capital of the West Ukrainian People's Republic. Between the wars, the city was the centre of the Lwów Voivodeship in the Se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Lwów
The University of Lviv ( uk, Львівський університет, Lvivskyi universytet; pl, Uniwersytet Lwowski; german: Universität Lemberg, briefly known as the ''Theresianum'' in the early 19th century), presently the Ivan Franko National University of Lviv ( uk, Львівський національний університет імені Івана Франка, Lvivskyi natsionalnyi universitet imeni Ivana Franka), is the oldest institution of higher learning in present-day Ukraine dating from 1661 when John II Casimir, King of Poland, granted it its first royal charter. Over the centuries, it has undergone various transformations, suspensions, and name changes that have reflected the geopolitical complexities of this part of Europe. The present institution can be dated to 1940. It is located in the historic city of Lviv in Lviv Oblast of Western Ukraine. History Polish–Lithuanian Commonwealth The university was founded on January 20, 1661, when King John ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach–Mazur Game
In general topology, set theory and game theory, a Banach– Mazur game is a topological game played by two players, trying to pin down elements in a set (space). The concept of a Banach–Mazur game is closely related to the concept of Baire spaces. This game was the first infinite positional game of perfect information to be studied. It was introduced by Stanisław Mazur as problem 43 in the Scottish book, and Mazur's questions about it were answered by Banach. Definition Let Y be a non-empty topological space, X a fixed subset of Y and \mathcal a family of subsets of Y that have the following properties: * Each member of \mathcal has non-empty interior. * Each non-empty open subset of Y contains a member of \mathcal. Players, P_1 and P_2 alternately choose elements from \mathcal to form a sequence W_0 \supseteq W_1 \supseteq \cdots. P_1 wins if and only if :X \cap \left (\bigcap_ W_n \right ) \neq \emptyset. Otherwise, P_2 wins. This is called a general Banach–Mazur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach–Mazur Theorem
In functional analysis, a field of mathematics, the Banach–Mazur theorem is a theorem roughly stating that most well-behaved normed spaces are subspaces of the space of continuous paths. It is named after Stefan Banach and Stanisław Mazur. Statement Every real, separable Banach space is isometrically isomorphic to a closed subspace of , the space of all continuous functions from the unit interval into the real line. Comments On the one hand, the Banach–Mazur theorem seems to tell us that the seemingly vast collection of all separable Banach spaces is not that vast or difficult to work with, since a separable Banach space is "only" a collection of continuous paths. On the other hand, the theorem tells us that is a "really big" space, big enough to contain every possible separable Banach space. Non-separable Banach spaces cannot embed isometrically in the separable space , but for every Banach space , one can find a compact Hausdorff space and an isometric linear e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Approximation Property
In mathematics, specifically functional analysis, a Banach space is said to have the approximation property (AP), if every compact operator is a limit of finite-rank operators. The converse is always true. Every Hilbert space has this property. There are, however, Banach spaces which do not; Per Enflo published the first counterexample in a 1973 article. However, much work in this area was done by Grothendieck (1955). Later many other counterexamples were found. The space of bounded operators on \ell^2 does not have the approximation property.Szankowski, A.B(H) does not have the approximation property.''Acta Math.'' 147, 89-108(1981). The spaces \ell^p for p\neq 2 and c_0 (see Sequence space) have closed subspaces that do not have the approximation property. Definition A locally convex topological vector space ''X'' is said to have the approximation property, if the identity map can be approximated, uniformly on precompact sets, by continuous linear maps of finite rank. Fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Warsaw
The University of Warsaw ( pl, Uniwersytet Warszawski, la, Universitas Varsoviensis) is a public university in Warsaw, Poland. Established in 1816, it is the largest institution of higher learning in the country offering 37 different fields of study as well as 100 specializations in humanities, technical, and the natural sciences. The University of Warsaw consists of 126 buildings and educational complexes with over 18 faculties: biology, chemistry, journalism and political science, philosophy and sociology, physics, geography and regional studies, geology, history, applied linguistics and philology, Polish language, pedagogy, economics, law and public administration, psychology, applied social sciences, management and mathematics, computer science and mechanics. The University of Warsaw is one of the top Polish universities. It was ranked by '' Perspektywy'' magazine as best Polish university in 2010, 2011, 2014, and 2016. International rankings such as ARW ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Per Enflo
Per H. Enflo (; born 20 May 1944) is a Swedish mathematician working primarily in functional analysis, a field in which he solved mathematical problems, problems that had been considered fundamental. Three of these problems had been open problem, open for more than forty years: * The Schauder basis#Basis problem, basis problem and the Approximation property, approximation problem and later * the invariant subspace problem for Banach spaces. In solving these problems, Enflo developed new techniques which were then used by other researchers in functional analysis and operator theory for years. Some of Enflo's research has been important also in other mathematical fields, such as number theory, and in computer science, especially computer algebra and approximation algorithms. Enflo works at Kent State University, where he holds the title of University Professor. Enflo has earlier held positions at the Miller Institute for Basic Research in Science at the University of California, Ber ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schauder Basis
In mathematics, a Schauder basis or countable basis is similar to the usual ( Hamel) basis of a vector space; the difference is that Hamel bases use linear combinations that are finite sums, while for Schauder bases they may be infinite sums. This makes Schauder bases more suitable for the analysis of infinite-dimensional topological vector spaces including Banach spaces. Schauder bases were described by Juliusz Schauder in 1927, although such bases were discussed earlier. For example, the Haar basis was given in 1909, and Georg Faber discussed in 1910 a basis for continuous functions on an interval, sometimes called a Faber–Schauder system.Faber, Georg (1910), "Über die Orthogonalfunktionen des Herrn Haar", ''Deutsche Math.-Ver'' (in German) 19: 104–112. ; http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN37721857X ; http://resolver.sub.uni-goettingen.de/purl?GDZPPN002122553 Definitions Let ''V'' denote a topological vector space over the field ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the 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 and Eduard Helly. Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term " Fréchet space." Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces. Definition A Banach space is a complete n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Approximation Problem
In functional analysis, a branch of mathematics, a compact operator is a linear operator T: X \to Y, where X,Y are normed vector spaces, with the property that T maps bounded subsets of X to relatively compact subsets of Y (subsets with compact closure in Y). Such an operator is necessarily a bounded operator, and so continuous. Some authors require that X,Y are Banach, but the definition can be extended to more general spaces. Any bounded operator ''T'' that has finite rank is a compact operator; indeed, the class of compact operators is a natural generalization of the class of finite-rank operators in an infinite-dimensional setting. When ''Y'' is a Hilbert space, it is true that any compact operator is a limit of finite-rank operators, so that the class of compact operators can be defined alternatively as the closure of the set of finite-rank operators in the norm topology. Whether this was true in general for Banach spaces (the approximation property) was an unsolved questio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scottish Café
The Scottish Café ( pl, Kawiarnia Szkocka) was a café in Lwów, Poland (now Lviv, Ukraine) where, in the 1930s and 1940s, mathematicians from the Lwów School of Mathematics collaboratively discussed research problems, particularly in functional analysis and topology. Stanisław Ulam recounts that the tables of the café had marble tops, so they could write in pencil, directly on the table, during their discussions. To keep the results from being lost, and after becoming annoyed with their writing directly on the table tops, Stefan Banach's wife provided the mathematicians with a large notebook, which was used for writing the problems and answers and eventually became known as the '' Scottish Book''. The book—a collection of solved, unsolved, and even probably unsolvable problems—could be borrowed by any of the guests of the café. Solving any of the problems was rewarded with prizes, with the most difficult and challenging problems having expensive prizes (during the Gre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
