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Polar Set
In functional and convex analysis, and related disciplines of mathematics, the polar set A^ is a special convex set associated to any subset A of a vector space X lying in the dual space X^. The bipolar of a subset is the polar of A^, but lies in X (not X^). Definitions There are at least three competing definitions of the polar of a set, originating in projective geometry and convex analysis. In each case, the definition describes a duality between certain subsets of a pairing of vector spaces \langle X, Y \rangle over the real or complex numbers (X and Y are often topological vector spaces (TVSs)). If X is a vector space over the field \mathbb then unless indicated otherwise, Y will usually, but not always, be some vector space of linear functionals on X and the dual pairing \left\langle \cdot, \cdot \right\rangle : X \times Y \to \mathbb will be the bilinear () defined by :\left\langle x, f \right\rangle := f(x). If X is a topological vector space then the space Y w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, Topological space#Definition, topology, etc.) and the linear transformation, linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of function space, spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous function, continuous, unitary operator, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential equations, differential and integral equations. The usage of the word ''functional (mathematics), functional'' as a noun goes back to the calculus of variati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Length
Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (album), 1997 album by the Australian band Regurgitator * The Units, a synthpunk band Television * ''The Unit'', an American television series * '' The Unit: Idol Rebooting Project'', South Korean reality TV survival show Business * Stock keeping unit, a discrete inventory management construct * Strategic business unit, a profit center which focuses on product offering and market segment * Unit of account, a monetary unit of measurement * Unit coin, a small coin or medallion (usually military), bearing an organization's insignia or emblem * Work unit, the name given to a place of employment in the People's Republic of China Science and technology Science and medicine * Unit, a vessel or section of a chemical plant * Blood unit, a measuremen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i.e. that the space not exclude any ''limiting values'' of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval ,1would be compact. Similarly, the space of rational numbers \mathbb is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers \mathbb is not compact either, because it excludes the two limiting values +\infty and -\infty. However, the ''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topologic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lower bound'' (abbreviated as ) is also commonly used. The supremum (abbreviated sup; plural suprema) of a subset S of a partially ordered set P is the least element in P that is greater than or equal to each element of S, if such an element exists. Consequently, the supremum is also referred to as the ''least upper bound'' (or ). The infimum is in a precise sense dual to the concept of a supremum. Infima and suprema of real numbers are common special cases that are important in analysis, and especially in Lebesgue integration. However, the general definitions remain valid in the more abstract setting of order theory where arbitrary partially ordered sets are considered. The concepts of infimum and supremum are close to minimum and max ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Weak-* Topology
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a topological vector space (such as a normed vector space) with respect to its continuous dual. The remainder of this article will deal with this case, which is one of the concepts of functional analysis. One may call subsets of a topological vector space weakly closed (respectively, weakly compact, etc.) if they are closed (respectively, compact, etc.) with respect to the weak topology. Likewise, functions are sometimes called weakly continuous (respectively, weakly differentiable, weakly analytic, etc.) if they are continuous (respectively, differentiable, analytic, etc.) with respect to the weak topology. History Starting in the early 1900s, David Hilbert and Marcel Riesz made extensive use of weak convergence. The early pioneers of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach–Alaoglu Theorem
In functional analysis and related branches of mathematics, the Banach–Alaoglu theorem (also known as Alaoglu's theorem) states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. A common proof identifies the unit ball with the weak-* topology as a closed subset of a product of compact sets with the product topology. As a consequence of Tychonoff's theorem, this product, and hence the unit ball within, is compact. This theorem has applications in physics when one describes the set of states of an algebra of observables, namely that any state can be written as a convex linear combination of so-called pure states. History According to Lawrence Narici and Edward Beckenstein, the Alaoglu theorem is a "very important result - maybe most important fact about the weak-* topology - hatechos throughout functional analysis." In 1912, Helly proved that the unit ball of the continuous dual space of C( , b is countably weak-* compact. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closure (topology)
In topology, the closure of a subset of points in a topological space consists of all points in together with all limit points of . The closure of may equivalently be defined as the union of and its boundary, and also as the intersection of all closed sets containing . Intuitively, the closure can be thought of as all the points that are either in or "near" . A point which is in the closure of is a point of closure of . The notion of closure is in many ways dual to the notion of interior. Definitions Point of closure For S as a subset of a Euclidean space, x is a point of closure of S if every open ball centered at x contains a point of S (this point can be x itself). This definition generalizes to any subset S of a metric space X. Fully expressed, for X as a metric space with metric d, x is a point of closure of S if for every r > 0 there exists some s \in S such that the distance d(x, s) < r ( is allowed). Another way to express this is to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Topology
In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean distance, Euclidean metric. Definition The Euclidean norm on \R^n is the non-negative function \, \cdot\, : \R^n \to \R defined by \left\, \left(p_1, \ldots, p_n\right)\right\, ~:=~ \sqrt. Like all Norm (mathematics), norms, it induces a canonical Metric (mathematics), metric defined by d(p, q) = \, p - q\, . The metric d : \R^n \times \R^n \to \R induced by the Euclidean norm is called the Euclidean metric or the Euclidean distance and the distance between points p = \left(p_1, \ldots, p_n\right) and q = \left(q_1, \ldots, q_n\right) is d(p, q) ~=~ \, p - q\, ~=~ \sqrt. In any metric space, the Ball (mathematics), open balls form a Base (topology), base for a topology on that space.Metric space, Metric space#Open and closed sets.2C topology and convergence The Euclidean topology on \R^n is the topology by these balls ... [...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] |
Complete Topological Vector Space
In functional analysis and related areas of mathematics, a complete topological vector space is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point x towards which they all get closer. The notion of "points that get progressively closer" is made rigorous by or , which are generalizations of , while "point x towards which they all get closer" means that this Cauchy net or filter converges to x. The notion of completeness for TVSs uses the theory of uniform spaces as a framework to generalize the notion of completeness for metric spaces. But unlike metric-completeness, TVS-completeness does not depend on any metric and is defined for TVSs, including those that are not metrizable or Hausdorff. Completeness is an extremely important property for a topological vector space to possess. The notions of completeness for normed spaces and metrizable TVSs, which are commonly defined in terms of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T2) is the most frequently used and discussed. It implies the uniqueness of limits of sequences, nets, and filters. Hausdorff spaces are named after Felix Hausdorff, one of the founders of topology. Hausdorff's original definition of a topological space (in 1914) included the Hausdorff condition as an axiom. Definitions Points x and y in a topological space X can be '' separated by neighbourhoods'' if there exists a neighbourhood U of x and a neighbourhood V of y such that U and V are disjoint (U\cap V=\varnothing). X is a Hausdorff space if any two distinct points in X are separated by neighbourhoods. This condition is the third separation axiom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
