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Anderson–Kadec Theorem
In mathematics, in the areas of topology and functional analysis, the Anderson–Kadec theorem states that any two infinite-dimensional, separable Banach spaces, or, more generally, Fréchet spaces, are homeomorphic as topological spaces. The theorem was proved by Mikhail Kadec (1966) and Richard Davis Anderson. Statement Every infinite-dimensional, separable Fréchet space is homeomorphic to \R^, the Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ... of countably many copies of the real line \R. Preliminaries Kadec norm: A norm \, \,\cdot\,\, on a normed linear space X is called a '' with respect to a total subset A \subseteq X^*'' of the dual space X^* if for each sequence x_n\in X the following condition is satisfied: * If \lim_ x^*\left(x_n\right ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Product
In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product is taken, the cells of the table contain ordered pairs of the form . One can similarly define the Cartesian product of sets, also known as an -fold Cartesian product, which can be represented by an -dimensional array, where each element is an -tuple. An ordered pair is a 2-tuple or couple. More generally still, one can define the Cartesian product of an indexed family of sets. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to the concept, which is further generalized in terms of direct product. Set-theoretic definition A rigorous definition of the Cartesian product re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Vector Spaces
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is also a topological space with the property that the vector space operations (vector addition and scalar multiplication) are also continuous functions. Such a topology is called a and every topological vector space has a uniform topological structure, allowing a notion of uniform convergence and completeness. Some authors also require that the space is a Hausdorff space (although this article does not). One of the most widely studied categories of TVSs are locally convex topological vector spaces. This article focuses on TVSs that are not necessarily locally convex. Other well-known examples of TVSs include Banach spaces, Hilbert spaces and Sobolev spaces. Many topological vector spaces are spaces of functions, or linear operators actin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Meier Eidelheit
Meier "Maks" Eidelheit (6 July 1910 – March 1943) was a Polish mathematician belonging to the Lwów School of Mathematics who worked in Lwów and was murdered in the Holocaust. Biography Meier Eidelheit left the Lwów Gymnasium in 1929 and then studied mathematics at the scientific faculty in Lwów, completing his study in 1933 with a thesis on the theory of summation. In 1938, with Stefan Banach as supervisor, he gained a doctorate from the Jan-Kazimierz-Lviv University, University of Lwów with a Dissertation ''über die Auflösbarkeit eines linearen Gleichungssystems mit unendlich vielen Unbekannten''. From 1933 to 1939 he gave private lectures; from 31 January 1939 onwards he was an Assistant Professor of Mathematical analysis, Analysis, from 21 March 1941 he was candidate for a professorship. He worked mainly on Functional analysis. On the basis of his 1936 paper on convex sets in linear normed spaces, geometric versions of the hyperplane separation theorem are also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Set
In functional analysis, a total set (also called a complete set) in a vector space is a set of linear functionals T with the property that if a vector x \in X satisfies f(x) = 0 for all f \in T, then x = 0 is the zero vector. In a more general setting, a subset T of a topological vector space X is a total set or fundamental set if the linear span of T is dense Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be use ... in X. See also * * * * References {{linear-algebra-stub Linear algebra Topological vector spaces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normed Vector Space
The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk, France, in 1898. The shipyard boomed in the period before World War I (1914–18), but struggled in the inter-war period. It was badly damaged during World War II (1939–45). In the first thirty years after the war the shipyard again experienced a boom and employed up to 3,000 workers making oil tankers, and then liquid natural gas tankers. Demand dropped off in the 1970s and 1980s. In 1972 the shipyard became Chantiers de France-Dunkerque, and in 1983 merged with others yards to become part of Chantiers du Nord et de la Mediterranee, or Normed. The shipyard closed in 1987. Foundation (1898–99) The Ateliers et Chantiers de France (ACF) company was officially founded on 6 July 1898 by a consortium of six shipping brokers, the Dunkirk chamber of commerce and the state. The state asked that the shipyard be able to build steamships and also four-maste ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Norm (mathematics)
In mathematics, a norm is a function (mathematics), function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the Origin (mathematics), origin: it Equivariant map, commutes with scaling, obeys a form of the triangle inequality, and zero is only at the origin. In particular, the Euclidean distance in a Euclidean space is defined by a norm on the associated Euclidean vector space, called the #Euclidean norm, Euclidean norm, the #p-norm, 2-norm, or, sometimes, the magnitude or length of the vector. This norm can be defined as the square root of the inner product of a vector with itself. A seminorm satisfies the first two properties of a norm but may be zero for vectors other than the origin. A vector space with a specified norm is called a normed vector space. In a similar manner, a vector space with a seminorm is called a ''seminormed vector space''. The term pseudonorm has been used for several related meaning ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Countable Set
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements. In more technical terms, assuming the axiom of countable choice, a set is ''countable'' if its cardinality (the number of elements of the set) is not greater than that of the natural numbers. A countable set that is not finite is said to be countably infinite. The concept is attributed to Georg Cantor, who proved the existence of uncountable sets, that is, sets that are not countable; for example the set of the real numbers. A note on terminology Although the terms "countable" and "countably infinite" as def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Davis Anderson
Richard Davis Anderson Sr. (February 17, 1922 – March 4, 2008) was an American mathematician known internationally for his work in infinite-dimensional topology. Much of his early work focused on proofs surrounding Hilbert space and Hilbert cubes. __TOC__ Life Richard Anderson and his twin brother, John, were born February 17, 1922, in Hamden, Connecticut. He received a bachelor's degree in mathematics from the University of Minnesota in 1941, after just two years of study. He went on to graduate school at the University of Texas at Austin, where he studied under R. L. Moore. His graduate work was interrupted by World War II. Two days after the Japanese attack on Pearl Harbor, he enlisted in the United States Navy. During his term in the U. S. Navy, he served on the USS ''Rocky Mount''. After returning from the war, he finished his doctoral work at the University of Texas and went on to teach mathematics at the University of Pennsylvania, where he went through t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such as Stretch factor, stretching, Torsion (mechanics), twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a Set (mathematics), set endowed with a structure, called a ''Topology (structure), topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of List of continuity-related mathematical topics, continuity. Euclidean spaces, and, more generally, metric spaces are examples of topological spaces, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and Homotopy, homotopies. A property that is invariant under such deformations is a to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mikhail Kadets
Mikhail Iosiphovich Kadets (, , sometimes transliterated as Kadec, 30 November 1923 – 7 March 2011) was a Soviet-born Jewish mathematician working in analysis and the theory of Banach spaces. Life and work Kadets was born in Kyiv. In 1943, he was drafted into the army. After demobilisation in 1946, he studied at Kharkov University, graduating in 1950. After several years in Makeevka he returned to Kharkov in 1957, where he spent the remainder of his life working at various institutes. He defended his PhD in 1955 (under the supervision of Boris Levin), and his doctoral dissertation in 1963. He was awarded the State Prize of Ukraine in 2005. After reading the Ukrainian translation of Banach's monograph '' Théorie des Opérations Linéaires'', he became interested in the theory of Banach spaces. In 1966, Kadets solved in the affirmative the Banach– Fréchet problem, asking whether every two separable infinite-dimensional Banach spaces are homeomorphic. He developed t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homeomorphic
In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. Very roughly speaking, a topological space is a geometric object, and a homeomorphism results from a continuous deformation of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a torus are not. However, this description can be misleading. Some continuous deformations do not produce homeomorphisms, such as the deformation of a li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
