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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 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 In mathematics, more specifically in functional analysis, a subset T of a topological vector space X is said to be a total subset of X if the linear span of T is a dense subset of X. This condition arises frequently in many theorems of functional ... 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 an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. 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 ''n'' sets, also known as an ''n''-fold Cartesian product, which can be represented by an ''n''-dimensional array, where each element is an ''n''- 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. Examples A deck of cards A ... [...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. Banach spaces, Hilbert spaces and Sobolev spaces are other well-known examples of TVSs. Many topological vector spaces are spaces of functions, or linear operators acting on t ... [...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- 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 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 known (in German) as ''Trennungssatz vo ... [...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 substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ... in X. See also * * * * References {{linear-algebra-stub Linear algebra Topological vector spaces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normed Vector Space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" in the real (physical) world. A norm is a real-valued function defined on the vector space that is commonly denoted x\mapsto \, x\, , and has the following properties: #It is nonnegative, meaning that \, x\, \geq 0 for every vector x. #It is positive on nonzero vectors, that is, \, x\, = 0 \text x = 0. # For every vector x, and every scalar \alpha, \, \alpha x\, = , \alpha, \, \, x\, . # The triangle inequality holds; that is, for every vectors x and y, \, x+y\, \leq \, x\, + \, y\, . A norm induces a distance, called its , by the formula d(x,y) = \, y-x\, . which makes any normed vector space into a metric space and a topological vector space. If this metric space is complete then the normed space is a Banach space. Every normed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Norm (mathematics)
In mathematics, a norm is a 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: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance of a vector from the origin is a norm, called the Euclidean norm, or 2-norm, which may also 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 meanings. It may be a synonym of "seminorm". A pseudonorm may satisfy the same axioms as a norm, with the equality replaced by an inequality "\,\leq\," in the homogeneity axiom. It can als ... [...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 (its number of elements) is not greater than that of the natural numbers. A countable set that is not finite is said 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 defined here are ... [...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 throug ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set endowed with a structure, called a '' topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological property. Basic examples of topological properties are: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mikhail Kadets
Mikhail Iosiphovich Kadets (russian: Михаил Иосифович Кадец, uk, Михайло Йосипович Кадець , 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 Kiev. 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, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homeomorphic
In the mathematical field of topology, a homeomorphism, 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. The word ''homeomorphism'' comes from the Greek words '' ὅμοιος'' (''homoios'') = similar or same and '' μορφή'' (''morphē'') = shape or form, introduced to mathematics by Henri Poincaré in 1895. Very roughly speaking, a topological space is a geometric object, and the homeomorphism is a continuous stretching and bending 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
