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Dixmier–Ng Theorem
In functional analysis, the Dixmier–Ng theorem is a characterization of when a normed space is in fact a dual Banach space. It was proven by Kung-fu Ng, who called it a variant of a theorem proven earlier by Jacques Dixmier. : Dixmier-Ng theorem. Let X be a normed space. The following are equivalent: # There exists a Hausdorff locally convex topology \tau on X so that the closed unit ball, \mathbf_X, of X is \tau-compact. # There exists a Banach space Y so that X is isometrically isomorphic to the dual of Y. That 2. implies 1. is an application of the Banach–Alaoglu theorem, setting \tau to the Weak-* topology. That 1. implies 2. is an application of the Bipolar theorem. Applications Let M be a pointed metric space with distinguished point denoted 0_M. The Dixmier-Ng Theorem is applied to show that the Lipschitz space \text_0(M) of all real-valued Lipschitz functions from M to \mathbb that vanish at 0_M (endowed with the Lipschitz constant In mathematical analysis, Lip ... [...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 (for example, Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, or Topological space#Definitions, topology) and the linear transformation, linear functions defined on these spaces and suitably respecting these structures. 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, for example, continuous function, continuous or unitary operator, unitary 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 v ... [...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] |
Banach Space
In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term " Fréchet space". Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces. Definition A Banach space is a complete nor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacques Dixmier
Jacques Dixmier (born 24 May 1924) is a French mathematician. He worked on operator algebras, especially C*-algebras, and wrote several of the standard reference books on them, and introduced the Dixmier trace and the Dixmier mapping. Biography Dixmier received his Ph.D. in 1949 from the University of Paris, and his students include Alain Connes. In 1949 upon the initiative of Jean-Pierre Serre and Pierre Samuel, Dixmier became a member of Bourbaki, in which he made essential contributions to the Bourbaki volume on Lie algebras. After retiring as professor emeritus from the University of Paris VI, he spent five years at the Institut des Hautes Études Scientifiques. Often, there is made the erroneous claim that Dixmier originated the name ''von Neumann algebra'' for the operator algebras introduced by John von Neumann, but Dixmier said in an interview that the name originated from a proposal by Jean Dieudonné. Dixmier was an invited speaker at the International Congr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Space
In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. 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 (after T0 and T1), which is why Hausdorff ... [...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 topological vector 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Unit Ball
In mathematics, a unit sphere is a sphere of unit radius: the set of points at Euclidean distance 1 from some center point in three-dimensional space. More generally, the ''unit -sphere'' is an -sphere of unit radius in -dimensional Euclidean space; the unit circle is a special case, the unit -sphere in the plane. An (open) unit ball is the region inside of a unit sphere, the set of points of distance less than 1 from the center. A sphere or ball with unit radius and center at the origin of the space is called ''the'' unit sphere or ''the'' unit ball. Any arbitrary sphere can be transformed to the unit sphere by a combination of translation and scaling, so the study of spheres in general can often be reduced to the study of the unit sphere. The unit sphere is often used as a model for spherical geometry because it has constant sectional curvature of 1, which simplifies calculations. In trigonometry, circular arc length on the unit circle is called radians and used for mea ... [...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-* co ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipolar Theorem
In mathematics, the bipolar theorem is a theorem in functional analysis that characterizes the bipolar (that is, the polar of the polar) of a set. In convex analysis, the bipolar theorem refers to a necessary and sufficient conditions for a cone to be equal to its bipolar. The bipolar theorem can be seen as a special case of the Fenchel–Moreau theorem. Preliminaries Suppose that X is a topological vector space (TVS) with a continuous dual space X^ and let \left\langle x, x^ \right\rangle := x^(x) for all x \in X and x^ \in X^. The convex hull of a set A, denoted by \operatorname A, is the smallest convex set containing A. The convex balanced hull of a set A is the smallest convex balanced set containing A. The polar of a subset A \subseteq X is defined to be: A^\circ := \left\. while the prepolar of a subset B \subseteq X^ is: ^ B := \left\. The bipolar of a subset A \subseteq X, often denoted by A^ is the set A^ := ^\left(A^\right) = \left\. Statement in functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pointed Space
In mathematics, a pointed space or based space is a topological space with a distinguished point, the basepoint. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as x_0, that remains unchanged during subsequent discussion, and is kept track of during all operations. Maps of pointed spaces (based maps) are continuous maps preserving basepoints, i.e., a map f between a pointed space X with basepoint x_0 and a pointed space Y with basepoint y_0 is a based map if it is continuous with respect to the topologies of X and Y and if f\left(x_0\right) = y_0. This is usually denoted :f : \left(X, x_0\right) \to \left(Y, y_0\right). Pointed spaces are important in algebraic topology, particularly in homotopy theory, where many constructions, such as the fundamental group, depend on a choice of basepoint. The pointed set concept is less important; it is anyway the case of a pointed discrete space. Pointed spaces are often taken a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), function called a metric or distance function. Metric spaces are a general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a Conceptual metaphor , metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are a tool used in many different bra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
