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Refinement (category Theory)
In category theory and related fields of mathematics, a refinement is a construction that generalizes the operations of "interior enrichment", like bornologification or saturation of a locally convex space. A dual construction is called envelope. Definition Suppose K is a category, X an object in K, and \Gamma and \Phi two classes of morphisms in K. The definition of a refinement of X in the class \Gamma by means of the class \Phi consists of two steps. * A morphism \sigma:X'\to X in K is called an ''enrichment of the object X in the class of morphisms \Gamma by means of the class of morphisms \Phi'', if \sigma\in\Gamma, and for any morphism \varphi:B\to X from the class \Phi there exists a unique morphism \varphi':B\to X' in K such that \varphi=\sigma\circ\varphi'. * An enrichment \rho:E\to X of the object X in the class of morphisms \Gamma by means of the class of morphisms \Phi is called a ''refinement of X in \Gamma by means of \Phi'', if for any other enrichment \sigma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, category theory is used in almost all areas of mathematics, and in some areas of computer science. In particular, many constructions of new mathematical objects from previous ones, that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality. A category is formed by two sorts of objects: the objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. One often says that a morphism is an ''arrow'' that ''maps'' its source to its target. Morphisms can be ''composed'' if the target of the first morphism equals the source of the second one, and morphism com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Envelope (category Theory)
In :Category theory and related fields of mathematics, an envelope is a construction that generalizes the operations of "exterior completion", like completion of a locally convex space, or Stone–Čech compactification of a topological space. A dual construction is called refinement. Definition Suppose K is a category, X an object in K, and \Omega and \Phi two classes of morphisms in K. The definition of an envelope of X in the class \Omega with respect to the class \Phi consists of two steps. * A morphism \sigma:X\to X' in K is called an ''extension of the object X in the class of morphisms \Omega with respect to the class of morphisms \Phi'', if \sigma\in\Omega, and for any morphism \varphi:X\to B from the class \Phi there exists a unique morphism \varphi':X'\to B in K such that \varphi=\varphi'\circ\sigma. * An extension \rho:X\to E of the object X in the class of morphisms \Omega with respect to the class of morphisms \Phi is called an ''envelope of X in \Omega with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Refinement
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Refinement may refer to: Mathematics * Equilibrium refinement, the identification of actualized equilibria in game theory * Refinement of an equivalence relation, in mathematics ** Refinement (topology), the refinement of an open cover in mathematical topology * Refinement (category theory) Other uses * Refinement (computing), computer science approaches for designing correct computer programs and enabling their formal verification * Refining, a process of purification ** Refining (metallurgy) * Refinement (culture), a quality of cultural sophistication *Refinement (horse), a racehorse ridden by jockey Tony McCoy Sir Anthony Peter McCoy (born 4 May 1974), commonly known as AP McCoy or Tony McCoy, is a Northern Irish former National Hunt horse racing jockey. Based in Ireland and the UK, McCoy rode a record 4,358 winners, and was Champion Jockey a reco ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Convex 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normed Space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" in the real (physical) world. A norm is a real-valued function defined on the vector space that is commonly denoted x\mapsto \, x\, , and has the following properties: #It is nonnegative, meaning that \, x\, \geq 0 for every vector x. #It is positive on nonzero vectors, that is, \, x\, = 0 \text x = 0. # For every vector x, and every scalar \alpha, \, \alpha x\, = , \alpha, \, \, x\, . # The triangle inequality holds; that is, for every vectors x and y, \, x+y\, \leq \, x\, + \, y\, . A norm induces a distance, called its , by the formula d(x,y) = \, y-x\, . which makes any normed vector space into a metric space and a topological vector space. If this metric space is complete then the normed space is a Banach space. Every nor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Vector Space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is also a topological space with the property that the vector space operations (vector addition and scalar multiplication) are also continuous functions. Such a topology is called a and every topological vector space has a uniform topological structure, allowing a notion of uniform convergence and completeness. Some authors also require that the space is a Hausdorff space (although this article does not). One of the most widely studied categories of TVSs are locally convex topological vector spaces. This article focuses on TVSs that are not necessarily locally convex. Banach spaces, Hilbert spaces and Sobolev spaces are other well-known examples of TVSs. Many topological vector spaces are spaces of functions, or linear operators acting o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Totally Bounded Set
In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “size” (where the meaning of “size” depends on the structure of the ambient space). The term precompact (or pre-compact) is sometimes used with the same meaning, but precompact is also used to mean relatively compact. These definitions coincide for subsets of a complete metric space, but not in general. In metric spaces A metric space (M,d) is ''totally bounded'' if and only if for every real number \varepsilon > 0, there exists a finite collection of open balls in ''M'' of radius \varepsilon whose union contains . Equivalently, the metric space ''M'' is totally bounded if and only if for every \varepsilon >0, there exists a finite cover such that the radius of each element of the cover is at most \varepsilon. This is equival ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Net (mathematics)
In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence. In essence, a sequence is a function whose domain is the natural numbers. The codomain of this function is usually some topological space. The motivation for generalizing the notion of a sequence is that, in the context of topology, sequences do not fully encode all information about functions between topological spaces. In particular, the following two conditions are, in general, not equivalent for a map f between topological spaces X and Y: #The map f is continuous in the topological sense; #Given any point x in X, and any sequence in X converging to x, the composition of f with this sequence converges to f(x) (continuous in the sequential sense). While it is necessarily true that condition 1 implies condition 2 (The truth of the condition 1 ensures the truth of the conditions 2.), the reverse implication is not nec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smith Space
In functional analysis and related areas of mathematics, a Smith space is a complete compactly generated locally convex topological vector space X having a ''universal compact set'', i.e. a compact set K which absorbs every other compact set T\subseteq X (i.e. T\subseteq\lambda\cdot K for some \lambda>0). Smith spaces are named afterMarianne Ruth Freundlich Smith who introduced them as duals to Banach spaces in some versions of duality theory for topological vector spaces. All Smith spaces are stereotype and are in the stereotype duality relations with Banach spaces: :* for any Banach space X its stereotype dual spaceThe ''stereotype dual'' space to a locally convex space X is the space X^\star of all linear continuous functionals f:X\to\mathbb endowed with the topology of uniform convergence on totally bounded sets in X. X^\star is a Smith space, :* and vice versa, for any Smith space X its stereotype dual space X^\star is a Banach space. Smith spaces are special cases of Br ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, category theory is used in almost all areas of mathematics, and in some areas of computer science. In particular, many constructions of new mathematical objects from previous ones, that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality. A category is formed by two sorts of objects: the objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. One often says that a morphism is an ''arrow'' that ''maps'' its source to its target. Morphisms can be ''composed'' if the target of the first morphism equals the source of the second one, and morphism com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duality Theories
Duality may refer to: Mathematics * Duality (mathematics), a mathematical concept ** Dual (category theory), a formalization of mathematical duality ** Duality (optimization) ** Duality (order theory), a concept regarding binary relations ** Duality (projective geometry), general principle of projective geometry ** Duality principle (Boolean algebra), the extension of order-theoretic duality to Boolean algebras ** S-duality (homotopy theory) * Philosophy, logic, and psychology * Dualistic cosmology, a twofold division in several spiritual and religious worldviews * Dualism (philosophy of mind), where the body and mind are considered to be irreducibly distinct * De Morgan's laws, specifically the ability to generate the dual of any logical expression * Complementary duality of Carl Jung's functions and types in Socionics Science Electrical and mechanical * Duality (electrical circuits), regarding isomorphism of electrical circuits * Duality (mechanical engineering), regarding i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
