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Topologies On Spaces 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] |
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] |
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] |
Linear Span
In mathematics, the linear span (also called the linear hull or just span) of a set of vectors (from a vector space), denoted , pp. 29-30, §§ 2.5, 2.8 is defined as the set of all linear combinations of the vectors in . It can be characterized either as the intersection of all linear subspaces that contain , or as the smallest subspace containing . The linear span of a set of vectors is therefore a vector space itself. Spans can be generalized to matroids and modules. To express that a vector space is a linear span of a subset , one commonly uses the following phrases—either: spans , is a spanning set of , is spanned/generated by , or is a generator or generator set of . Definition Given a vector space over a field , the span of a set of vectors (not necessarily infinite) is defined to be the intersection of all subspaces of that contain . is referred to as the subspace ''spanned by'' , or by the vectors in . Conversely, is called a ''spanning set'' of , and we ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Saturated Family
In mathematics, specifically in functional analysis, a family \mathcal of subsets a topological vector space (TVS) X is said to be saturated if \mathcal contains a non-empty subset of X and if for every G \in \mathcal, the following conditions all hold: # \mathcal contains every subset of G; # the union of any finite collection of elements of \mathcal is an element of \mathcal; # for every scalar a, \mathcal contains aG; # the closed convex balanced hull of G belongs to \mathcal. Definitions If \mathcal is any collection of subsets of X then the smallest saturated family containing \mathcal is called the of \mathcal. The family \mathcal is said to X if the union \bigcup_ G is equal to X; it is if the linear span of this set is a dense subset of X. Examples The intersection of an arbitrary family of saturated families is a saturated family. Since the power set of X is saturated, any given non-empty family \mathcal of subsets of X containing at least one non-empty set, the s ... [...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] |
Absolutely Convex
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] |
Minkowski Sum
In geometry, the Minkowski sum (also known as dilation) of two sets of position vectors ''A'' and ''B'' in Euclidean space is formed by adding each vector in ''A'' to each vector in ''B'', i.e., the set : A + B = \. Analogously, the Minkowski difference (or geometric difference) is defined using the complement operation as : A - B = \left(A^c + (-B)\right)^c In general A - B \ne A + (-B). For instance, in a one-dimensional case A = 2, 2/math> and B = 1, 1/math> the Minkowski difference A - B = 1, 1/math>, whereas A + (-B) = A + B = 3, 3 In a two-dimensional case, Minkowski difference is closely related to erosion (morphology) in image processing. The concept is named for Hermann Minkowski. Example For example, if we have two sets ''A'' and ''B'', each consisting of three position vectors (informally, three points), representing the vertices of two triangles in \mathbb^2, with coordinates :A = \ and :B = \ then their Minkowski sum is :A + B = \ which comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Filter Base
In mathematics, a filter on a set X is a family \mathcal of subsets such that: # X \in \mathcal and \emptyset \notin \mathcal # if A\in \mathcal and B \in \mathcal, then A\cap B\in \mathcal # If A,B\subset X,A\in \mathcal, and A\subset B, then B\in \mathcal A filter on a set may be thought of as representing a "collection of large subsets". Filters appear in order, model theory, set theory, but can also be found in topology, from which they originate. The dual notion of a filter is an ideal. Filters were introduced by Henri Cartan in 1937 and as described in the article dedicated to filters in topology, they were subsequently used by Nicolas Bourbaki in their book ''Topologie Générale'' as an alternative to the related notion of a net developed in 1922 by E. H. Moore and Herman L. Smith. Order filters are generalizations of filters from sets to arbitrary partially ordered sets. Specifically, a filter on a set is just a proper order filter in the special case where the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Numbers
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number a+bi, is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Numbers
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers is denoted or \mathbb and is sometimes called "the reals". The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality. The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real numbers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by constants. The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the . When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called the ''continuous dual space''. Dual vector spaces find application in many branches of mathematics that use vector spaces, such as in tensor analysis with finite-dimensional vector spaces. When applied to vector spaces of functions (which are typically infinite-dimensional), dual spaces are used to describe measures, distributions, and Hilbert spaces. Consequently, the dual space is an important concept in functional analysis. Early terms for ''dual'' include ''polarer Raum'' ahn 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metrizable TVS
In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of locally convex metrizable TVS. Pseudometrics and metrics A pseudometric on a set X is a map d : X \times X \rarr \R satisfying the following properties: d(x, x) = 0 \text x \in X; Symmetry: d(x, y) = d(y, x) \text x, y \in X; Subadditivity: d(x, z) \leq d(x, y) + d(y, z) \text x, y, z \in X. A pseudometric is called a metric if it satisfies: Identity of indiscernibles: for all x, y \in X, if d(x, y) = 0 then x = y. Ultrapseudometric A pseudometric d on X is called a ultrapseudometric or a strong pseudometric if it satisfies: Strong/Ultrametric triangle inequality: d(x, z) \leq \max \ \text x, y, z \in X. Pseudometric space A pseudometric space is a pair (X, d) consisting of a set X and a pseudometric d on X such that X's to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
