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Bidual
In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded subsets of X, where this topology is denoted by b\left(X^, X\right) or \beta\left(X^, X\right). The coarsest polar topology is called weak topology. The strong dual space plays such an important role in modern functional analysis, that the continuous dual space is usually assumed to have the strong dual topology unless indicated otherwise. To emphasize that the continuous dual space, X^, has the strong dual topology, X^_b or X^_ may be written. Strong dual topology Throughout, all vector spaces will be assumed to be over the field \mathbb of either the real numbers \R or complex numbers \C. Definition from a dual system Let (X, Y, \langle \cdot, \cdot \rangle) be a dual pair of vector spaces over the field \mathbb of real numbers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) 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 norme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Pair
In mathematics, a dual system, dual pair, or duality over a field \mathbb is a triple (X, Y, b) consisting of two vector spaces X and Y over \mathbb and a non-degenerate bilinear map b : X \times Y \to \mathbb. Duality theory, the study of dual systems, is part of functional analysis. According to Helmut H. Schaefer, "the study of a locally convex space in terms of its dual is the central part of the modern theory of topological vector spaces, for it provides the deepest and most beautiful results of the subject." Definition, notation, and conventions ;Pairings A or pair over a field \mathbb is a triple (X, Y, b), which may also be denoted by b(X, Y), consisting of two vector spaces X and Y over \mathbb (which this article assumes is either the real numbers or the complex numbers \Complex) and a bilinear map b : X \times Y \to \mathbb, which is called the bilinear map associated with the pairing or simply the pairing's map/bilinear form. For every x \in X, define \begin ... [...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] |
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] |
Metrizable Topological Vector Space
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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infrabarreled Space
In functional analysis, a discipline within mathematics, a locally convex topological vector space (TVS) is said to be infrabarrelled (also spelled infra barreled) if every bounded absorbing barrel is a neighborhood of the origin. Characterizations If X is a Hausdorff locally convex space then the canonical injection from X into its bidual is a topological embedding if and only if X is infrabarrelled. Properties Every quasi-complete infrabarrelled space is barrelled. Examples Every barrelled space In functional analysis and related areas of mathematics, a barrelled space (also written barreled space) is a topological vector space (TVS) for which every barrelled set in the space is a neighbourhood for the zero vector. A barrelled set or a ... is infrabarrelled. A closed vector subspace of an infrabarrelled space is, however, not necessarily infrabarrelled. Every product and locally convex direct sum of any family of infrabarrelled spaces is infrabarrelled. Eve ... [...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] |
LF-space
In mathematics, an ''LF''-space, also written (''LF'')-space, is a topological vector space (TVS) ''X'' that is a locally convex inductive limit of a countable inductive system (X_n, i_) of Fréchet spaces. This means that ''X'' is a direct limit of a direct system (X_n, i_) in the category of locally convex topological vector spaces and each X_n is a Fréchet space. The name ''LF'' stands for Limit of Fréchet spaces. If each of the bonding maps i_ is an embedding of TVSs then the ''LF''-space is called a strict ''LF''-space. This means that the subspace topology induced on by is identical to the original topology on . Some authors (e.g. Schaefer) define the term "''LF''-space" to mean "strict ''LF''-space," so when reading mathematical literature, it is recommended to always check how ''LF''-space is defined. Definition Inductive/final/direct limit topology Throughout, it is assumed that * \mathcal is either the category of topological spaces or some subcategory of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metrizable
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) such that the topology induced by d is \mathcal. Metrization theorems are theorems that give sufficient conditions for a topological space to be metrizable. Properties Metrizable spaces inherit all topological properties from metric spaces. For example, they are Hausdorff paracompact spaces (and hence normal and Tychonoff) and first-countable. However, some properties of the metric, such as completeness, cannot be said to be inherited. This is also true of other structures linked to the metric. A metrizable uniform space, for example, may have a different set of contraction maps than a metric space to which it is homeomorphic. Metrization theorems One of the first widely recognized metrization theorems was . This states that every H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Mackey Topology
In functional analysis and related areas of mathematics, the Mackey topology, named after George Mackey, is the finest topology for a topological vector space which still preserves the continuous dual. In other words the Mackey topology does not make linear functions continuous which were discontinuous in the default topology. A topological vector space (TVS) is called a Mackey space if its topology is the same as the Mackey topology. The Mackey topology is the opposite of the weak topology, which is the coarsest topology on a topological vector space which preserves the continuity of all linear functions in the continuous dual. The Mackey–Arens theorem states that all possible dual topologies are finer than the weak topology and coarser than the Mackey topology. Definition Definition for a pairing Given a pairing (X, Y, b), the Mackey topology on X induced by (X, Y, b), denoted by \tau(X, Y, b), is the polar topology defined on X by using the set of all \sigma(Y, X, b)-comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barreled Space
In functional analysis and related areas of mathematics, a barrelled space (also written barreled space) is a topological vector space (TVS) for which every barrelled set in the space is a neighbourhood for the zero vector. A barrelled set or a barrel in a topological vector space is a set that is convex, balanced, absorbing, and closed. Barrelled spaces are studied because a form of the Banach–Steinhaus theorem still holds for them. Barrelled spaces were introduced by . Barrels A convex and balanced subset of a real or complex vector space is called a and it is said to be , , or . A or a in a topological vector space (TVS) is a subset that is a closed absorbing disk; that is, a barrel is a convex, balanced, closed, and absorbing subset. Every barrel must contain the origin. If \dim X \geq 2 and if S is any subset of X, then S is a convex, balanced, and absorbing set of X if and only if this is all true of S \cap Y in Y for every 2-dimensional vector subspace Y; ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
