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Vector Bornology
In mathematics, especially functional analysis, a bornology \mathcal on a vector space X over a field \mathbb, where \mathbb has a bornology ℬ\mathbb, is called a vector bornology if \mathcal makes the vector space operations into bounded maps. Definitions Prerequisits A on a set X is a collection \mathcal of subsets of X that satisfy all the following conditions: #\mathcal covers X; that is, X = \cup \mathcal #\mathcal is stable under inclusions; that is, if B \in \mathcal and A \subseteq B, then A \in \mathcal #\mathcal is stable under finite unions; that is, if B_1, \ldots, B_n \in \mathcal then B_1 \cup \cdots \cup B_n \in \mathcal Elements of the collection \mathcal are called or simply if \mathcal is understood. The pair (X, \mathcal) is called a or a . A or of a bornology \mathcal is a subset \mathcal_0 of \mathcal such that each element of \mathcal is a subset of some element of \mathcal_0. Given a collection \mathcal of subsets of X, the smallest bornolog ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolutely Convex Set
In mathematics, a subset ''C'' of a real or complex vector space is said to be absolutely convex or disked if it is convex and balanced (some people use the term "circled" instead of "balanced"), in which case it is called a disk. The disked hull or the absolute convex hull of a set is the intersection of all disks containing that set. Definition A subset S of a real or complex vector space X is called a ' and is said to be ', ', and ' if any of the following equivalent conditions is satisfied: S is a convex and balanced set. for any scalar a and b, if , a, + , b, \leq 1 then a S + b S \subseteq S. for all scalars a, b, and c, if , a, + , b, \leq , c, , then a S + b S \subseteq c S. for any scalars a_1, \ldots, a_n and c, if , a_1, + \cdots + , a_n, \leq , c, then a_1 S + \cdots + a_n S \subseteq c S. for any scalars a_1, \ldots, a_n, if , a_1, + \cdots + , a_n, \leq 1 then a_1 S + \cdots + a_n S \subseteq S. The smallest convex (respectively, balanced) subset o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ultrabornological Space
In functional analysis, a topological vector space (TVS) X is called ultrabornological if every bounded linear operator from X into another TVS is necessarily continuous. A general version of the closed graph theorem holds for ultrabornological spaces. Ultrabornological spaces were introduced by Alexander Grothendieck (Grothendieck 955, p. 17"espace du type (β)"). Definitions Let X be a topological vector space (TVS). Preliminaries A disk is a convex and balanced set. A disk in a TVS X is called bornivorous if it absorbs every bounded subset of X. A linear map between two TVSs is called infrabounded if it maps Banach disks to bounded disks. A disk D in a TVS X is called infrabornivorous if it satisfies any of the following equivalent conditions: D absorbs every Banach disks in X. while if X locally convex then we may add to this list: the gauge of D is an infrabounded map; while if X locally convex and Hausdorff then we may add to this list: D absorbs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Space Of Linear Maps
In mathematics, particularly functional analysis, spaces of linear maps between two vector spaces can be endowed with a variety of topologies. Studying space of linear maps and these topologies can give insight into the spaces themselves. The article operator topologies discusses topologies on spaces of linear maps between normed spaces, whereas this article discusses topologies on such spaces in the more general setting of topological vector spaces (TVSs). Topologies of uniform convergence on arbitrary spaces of maps Throughout, the following is assumed: T is any non-empty set and \mathcal is a non-empty collection of subsets of T directed by subset inclusion (i.e. for any G, H \in \mathcal there exists some K \in \mathcal such that G \cup H \subseteq K). Y is a topological vector space (not necessarily Hausdorff or locally convex). \mathcal is a basis of neighborhoods of 0 in Y. F is a vector subspace of Y^T = \prod_ Y,Because T is just a set that is not yet assumed to be endo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bornology
In mathematics, especially functional analysis, a bornology on a set ''X'' is a collection of subsets of ''X'' satisfying axioms that generalize the notion of boundedness. One of the key motivations behind bornologies and bornological analysis is the fact that bornological spaces provide a convenient setting for homological algebra in functional analysis. This is becausepg 9 the category of bornological spaces is additive, complete, cocomplete, and has a tensor product adjoint to an internal hom, all necessary components for homological algebra. History Bornology originates from functional analysis. There are two natural ways of studying the problems of functional analysis: one way is to study notions related to topologies ( vector topologies, continuous operators, open/compact subsets, etc.) and the other is to study notions related to boundedness ( vector bornologies, bounded operators, bounded subsets, etc.). For normed spaces, from which functional analysis arose, the disti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bornological Space
In mathematics, particularly in functional analysis, a bornological space is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of sets and linear maps, in the same way that a topological space possesses the minimum amount of structure needed to address questions of continuity. Bornological spaces are distinguished by the property that a linear map from a bornological space into any locally convex spaces is continuous if and only if it is a bounded linear operator. Bornological spaces were first studied by George Mackey. The name was coined by Bourbaki after , the French word for " bounded". Bornologies and bounded maps A on a set X is a collection \mathcal of subsets of X that satisfy all the following conditions: \mathcal covers X; that is, X = \cup \mathcal; \mathcal is stable under inclusions; that is, if B \in \mathcal and A \subseteq B, then A \in \mathcal; \mathcal is stable under finite unions; ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equicontinuous
In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In particular, the concept applies to countable families, and thus ''sequences'' of functions. Equicontinuity appears in the formulation of Ascoli's theorem, which states that a subset of ''C''(''X''), the space of continuous functions on a compact Hausdorff space ''X'', is compact if and only if it is closed, pointwise bounded and equicontinuous. As a corollary, a sequence in ''C''(''X'') is uniformly convergent if and only if it is equicontinuous and converges pointwise to a function (not necessarily continuous a-priori). In particular, the limit of an equicontinuous pointwise convergent sequence of continuous functions ''fn'' on either metric space or locally compact space is continuous. If, in addition, ''fn'' are holomorphic, then the limit is also holomorphic. The uniform bounde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Space
In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary). For instance, the set of rational numbers is not complete, because e.g. \sqrt is "missing" from it, even though one can construct a Cauchy sequence of rational numbers that converges to it (see further examples below). It is always possible to "fill all the holes", leading to the ''completion'' of a given space, as explained below. Definition Cauchy sequence A sequence x_1, x_2, x_3, \ldots in a metric space (X, d) is called Cauchy if for every positive real number r > 0 there is a positive integer N such that for all positive integers m, n > N, d\left(x_m, x_n\right) < r. Complete space A metric space is complete if any of the following equivalent conditions are satisfied: :#Every |
Seminorm
In mathematics, particularly in functional analysis, a seminorm is a vector space norm that 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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Convex Topological Vector Space
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals. Fréchet spaces are locally convex spaces that are completely metrizable (with a choice of complete metric). They are generalizations of Banach spaces, which are complete vector spaces with respect to a metric generated by a norm. History Metrizable topologies on vecto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bornivorous Set
In functional analysis, a subset of a real or complex vector space X that has an associated vector bornology \mathcal is called bornivorous and a bornivore if it absorbs every element of \mathcal. If X is a topological vector space (TVS) then a subset S of X is bornivorous if it is bornivorous with respect to the von-Neumann bornology of X. Bornivorous sets play an important role in the definitions of many classes of topological vector spaces, particularly bornological spaces. Definitions If X is a TVS then a subset S of X is called and a if S absorbs every bounded subset of X. An absorbing disk in a locally convex space is bornivorous if and only if its Minkowski functional is locally bounded (i.e. maps bounded sets to bounded sets). Infrabornivorous sets and infrabounded maps A linear map between two TVSs is called if it maps Banach disks to bounded disks. A disk in X is called if it absorbs every Banach disk. An absorbing disk in a locally conv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Convex
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals. Fréchet spaces are locally convex spaces that are completely metrizable (with a choice of complete metric). They are generalizations of Banach spaces, which are complete vector spaces with respect to a metric generated by a norm. History Metrizable topologies on vect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
