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Seminormed Space
In mathematics, particularly in functional analysis, a seminorm is like a norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and, conversely, the Minkowski functional of any such set is a seminorm. A topological vector space is locally convex if and only if its topology is induced by a family of seminorms. Definition Let X be a vector space over either the real numbers \R or the complex numbers \Complex. A real-valued function p : X \to \R is called a if it satisfies the following two conditions: # Subadditivity/Triangle inequality: p(x + y) \leq p(x) + p(y) for all x, y \in X. # Absolute homogeneity: p(s x) =, s, p(x) for all x \in X and all scalars s. These two conditions imply that p(0) = 0If z \in X denotes the zero vector in X while 0 denote the zero scalar, then absolute homogeneity implies that p(z) = p(0 z) = , 0, p(z) = 0 p(z) = 0. \blacksquare and that ev ... [...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] |
Normed 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-masted ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism. If a linear map is a bijection then it is called a . In the case where V = W, a linear map is called a linear endomorphism. Sometimes the term refers to this case, but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that V and W are real vector spaces (not necessarily with V = W), or it can be used to emphasize that V is a function space, which is a common convention in functional analysis. Sometimes the term ''linear function'' has the same meaning as ''linear m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distributive Lattice
In mathematics, a distributive lattice is a lattice (order), lattice in which the operations of join and meet distributivity, distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union (set theory), union and intersection (set theory), intersection. Indeed, these lattices of sets describe the scenery completely: every distributive lattice is—up to order isomorphism, isomorphism—given as such a lattice of sets. Definition As in the case of arbitrary lattices, one can choose to consider a distributive lattice ''L'' either as a structure of order theory or of universal algebra. Both views and their mutual correspondence are discussed in the article on lattice (order), lattices. In the present situation, the algebraic description appears to be more convenient. A lattice (''L'',∨,∧) is distributive if the following additional identity holds for all ''x'', ''y'', and ''z'' i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Subspace
Vector most often refers to: * Euclidean vector, a quantity with a magnitude and a direction * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics and physics * Vector (mathematics and physics) ** Row and column vectors, single row or column matrices ** Vector quantity ** Vector space ** Vector field, a vector for each point Molecular biology * Vector (molecular biology), a DNA molecule used as a vehicle to artificially carry foreign genetic material into another cell ** Cloning vector, a small piece of DNA into which a foreign DNA fragment can be inserted for cloning purposes ** Shuttle vector, a plasmid constructed so that it can propagate in two different host species ** Viral vector, a tool commonly used by molecular biologists to deliver genetic materials into cells Computer science * Vector, a one-dimensional array data structure ** Distance-vector routing protocol, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolute Value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), and For example, the absolute value of 3 and the absolute value of −3 is The absolute value of a number may be thought of as its distance from zero. Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces. The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts. Terminology and notation In 1806, Jean-Robert Argand introduced the term ''module'', meaning ''unit of measure'' in French, specifically for the ''complex'' absolute value,Oxford English Dictionary, Draft Revision, Ju ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Form
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear mapIn some texts the roles are reversed and vectors are defined as linear maps from covectors to scalars from a vector space to its field (mathematics), field of scalar (mathematics), scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the set of all linear functionals from to is itself a vector space over with addition and scalar multiplication defined pointwise. This space is called the dual space of , or sometimes the algebraic dual space, when a topological dual space is also considered. It is often denoted , p. 19, §3.1 or, when the field is understood, V^*; other notations are also used, such as V', V^ or V^. When vectors are represented by column vectors (as is common when a basis (linear algebra), basis is fixed), then linear functionals are represented as row vectors, and their values on specific vectors are given by matri ... [...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 for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the 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. Preliminaries The -norm in finite dimensions The Euclidean length of a vector x = (x_1, x_2, \dots, x_n) in the n-dimensional real vector space \Reals^n is given by the Euclidean norm: \, x\, _2 = \left(^2 + ^2 + \dotsb + ^2\right)^. The Euclidean distance between two points x and y is the length \, x - y\, _2 of the straight line b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lebesgue Measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it coincides with the standard measure of length, area, or volume. In general, it is also called '-dimensional volume, '-volume, hypervolume, or simply volume. It is used throughout real analysis, in particular to define Lebesgue integration. Sets that can be assigned a Lebesgue measure are called Lebesgue-measurable; the measure of the Lebesgue-measurable set A is here denoted by \lambda(A). Henri Lebesgue described this measure in the year 1901 which, a year after, was followed up by his description of the Lebesgue integral. Both were published as part of his dissertation in 1902. Definition For any interval I = ,b/math>, or I = (a, b), in the set \mathbb of real numbers, let \ell(I)= b - a denote its length. For any subset E\subseteq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indiscrete Topology
In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the consequence that all points of the space are "lumped together" and cannot be distinguished by topological means. Every indiscrete space can be viewed as a pseudometric space in which the distance between any two points is zero. Details The trivial topology is the topology with the least possible number of open sets, namely the empty set and the entire space, since the definition of a topology requires these two sets to be open. Despite its simplicity, a space ''X'' with more than one element and the trivial topology lacks a key desirable property: it is not a T0 space. Other properties of an indiscrete space ''X''—many of which are quite unusual—include: * The only closed sets are the empty set and ''X''. * The only possible basi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Balanced Function
In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field \mathbb with an absolute value function , \cdot , ) is a set S such that a S \subseteq S for all scalars a satisfying , a, \leq 1. The balanced hull or balanced envelope of a set S is the smallest balanced set containing S. The balanced core of a set S is the largest balanced set contained in S. Balanced sets are ubiquitous in functional analysis because every neighborhood of the origin in every topological vector space (TVS) contains a balanced neighborhood of the origin and every convex neighborhood of the origin contains a balanced convex neighborhood of the origin (even if the TVS is not locally convex). This neighborhood can also be chosen to be an open set or, alternatively, a closed set. Definition Let X be a vector space over the field \mathbb of real or complex numbers. Notation If S is a set, a is a scalar, and B \subseteq \mathbb then let ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sublinear Function
In linear algebra, a sublinear function (or functional as is more often used in functional analysis), also called a quasi-seminorm or a Banach functional, on a vector space X is a real-valued function with only some of the properties of a seminorm. Unlike seminorms, a sublinear function does not have to be nonnegative-valued and also does not have to be absolutely homogeneous. Seminorms are themselves abstractions of the more well known notion of norms, where a seminorm has all the defining properties of a norm that it is not required to map non-zero vectors to non-zero values. In functional analysis the name Banach functional is sometimes used, reflecting that they are most commonly used when applying a general formulation of the Hahn–Banach theorem. The notion of a sublinear function was introduced by Stefan Banach when he proved his version of the Hahn-Banach theorem. There is also a different notion in computer science, described below, that also goes by the name ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
