|
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] |
Vector Bornology
In mathematics, especially functional analysis, a bornology \mathcal on a vector space X over a field \mathbb, where \mathbb has a bornology ℬ\mathbb, is called a vector bornology if \mathcal makes the vector space operations into bounded maps. Definitions Prerequisits 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; that is, if B_1, \ldots, B_n \in \mathcal then B_1 \cup \cdots \cup B_n \in \mathcal Elements of the collection \mathcal are called or simply if \mathcal is understood. The pair (X, \mathcal) is called a or a . A or of a bornology \mathcal is a subset \mathcal_0 of \mathcal such that each element of \mathcal is a subset of some element of \mathcal_0. Given a collection \mathcal of subsets of X, the smallest bornolog ... [...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] |
Bounded Set (functional Analysis)
In functional analysis and related areas of mathematics, a set in a topological vector space is called bounded or von Neumann bounded, if every neighborhood of the zero vector can be ''inflated'' to include the set. A set that is not bounded is called unbounded. Bounded sets are a natural way to define locally convex polar topologies on the vector spaces in a dual pair, as the polar set of a bounded set is an absolutely convex and absorbing set. The concept was first introduced by John von Neumann and Andrey Kolmogorov in 1935. Definition Suppose X is a topological vector space (TVS) over a field \mathbb. A subset B of X is called or just in X if any of the following equivalent conditions are satisfied: : For every neighborhood V of the origin there exists a real r > 0 such that B \subseteq s VFor any set A and scalar s, the notation s A is denotes the set s A := \. for all scalars s satisfying , s, \geq r. * This was the definition introduced by John von Neumann in 193 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bounded Set (topological Vector Space)
In functional analysis and related areas of mathematics, a set in a topological vector space is called bounded or von Neumann bounded, if every neighborhood of the zero vector can be ''inflated'' to include the set. A set that is not bounded is called unbounded. Bounded sets are a natural way to define locally convex polar topologies on the vector spaces in a dual pair, as the polar set of a bounded set is an absolutely convex and absorbing set. The concept was first introduced by John von Neumann and Andrey Kolmogorov in 1935. Definition Suppose X is a topological vector space (TVS) over a field \mathbb. A subset B of X is called or just in X if any of the following equivalent conditions are satisfied: : For every neighborhood V of the origin there exists a real r > 0 such that B \subseteq s VFor any set A and scalar s, the notation s A is denotes the set s A := \. for all scalars s satisfying , s, \geq r. * This was the definition introduced by John von Neumann ... [...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] |
Mackey Space
In mathematics, particularly in functional analysis, a Mackey space is a locally convex topological vector space ''X'' such that the topology of ''X'' coincides with the Mackey topology τ(''X'',''X′''), the finest topology which still preserves the continuous dual. They are named after George Mackey. Examples Examples of locally convex spaces that are Mackey spaces include: * All barrelled spaces and more generally all infrabarreled spaces ** Hence in particular all bornological spaces and reflexive spaces * All metrizable spaces. ** In particular, all Fréchet spaces, including all Banach spaces and specifically Hilbert spaces, are Mackey spaces. * The product, locally convex direct sum, and the inductive limit of a family of Mackey spaces is a Mackey space.Schaefer (1999) p. 138 Properties * A locally convex space X with continuous dual X' is a Mackey space if and only if each convex and \sigma(X', X)-relatively compact subset of X' is equicontinuous. * The comple ... [...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] |
Mackey–Arens Theorem
The Mackey–Arens theorem is an important theorem in functional analysis that characterizes those locally convex vector topologies that have some given space of linear functionals as their continuous dual space. According to Narici (2011), this profound result is central to duality theory; a theory that is "the central part of the modern theory of topological vector spaces." Prerequisites Let be a vector space and let be a vector subspace of the algebraic dual of that separates points on . If is any other locally convex Hausdorff topological vector space topology on , then we say that is compatible with duality between and if when is equipped with , then it has as its continuous dual space. If we give the weak topology then is a Hausdorff locally convex topological vector space (TVS) and is compatible with duality between and (i.e. X_^ = \left( X_ \right)^ = Y). We can now ask the question: what are ''all'' of the locally convex Hausdorff TVS topologies that w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Algebra
In mathematics, a topological algebra A is an algebra and at the same time a topological space, where the algebraic and the topological structures are coherent in a specified sense. Definition A topological algebra A over a topological field K is a topological vector space together with a bilinear multiplication :\cdot: A \times A \to A, :(a,b) \mapsto a \cdot b that turns A into an algebra over K and is continuous in some definite sense. Usually the ''continuity of the multiplication'' is expressed by one of the following (non-equivalent) requirements: * ''joint continuity'': for each neighbourhood of zero U\subseteq A there are neighbourhoods of zero V\subseteq A and W\subseteq A such that V \cdot W\subseteq U (in other words, this condition means that the multiplication is continuous as a map between topological spaces or * ''stereotype continuity'': for each totally bounded set S\subseteq A and for each neighbourhood of zero U\subseteq A there is a neighbourhood of zero V\su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gelfand Representation
In mathematics, the Gelfand representation in functional analysis (named after I. M. Gelfand) is either of two things: * a way of representing commutative Banach algebras as algebras of continuous functions; * the fact that for commutative C*-algebras, this representation is an isometric isomorphism. In the former case, one may regard the Gelfand representation as a far-reaching generalization of the Fourier transform of an integrable function. In the latter case, the Gelfand–Naimark representation theorem is one avenue in the development of spectral theory for normal operators, and generalizes the notion of diagonalizing a normal matrix. Historical remarks One of Gelfand's original applications (and one which historically motivated much of the study of Banach algebras) was to give a much shorter and more conceptual proof of a celebrated lemma of Norbert Wiener (see the citation below), characterizing the elements of the group algebras ''L''1(R) and \ell^1() whose translates ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distribution (mathematics)
Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions than classical solutions, or where appropriate classical solutions may not exist. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are singular, such as the Dirac delta function. A function f is normally thought of as on the in the function domain by "sending" a point x in its domain to the point f(x). Instead of acting on points, distribution theory reinterpr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Calculus
In mathematics, a functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. It is now a branch (more accurately, several related areas) of the field of functional analysis, connected with spectral theory. (Historically, the term was also used synonymously with calculus of variations; this usage is obsolete, except for functional derivative. Sometimes it is used in relation to types of functional equations, or in logic for systems of predicate calculus.) If f is a function, say a numerical function of a real number, and M is an operator, there is no particular reason why the expression f(M) should make sense. If it does, then we are no longer using f on its original function domain. In the tradition of operational calculus, algebraic expressions in operators are handled irrespective of their meaning. This passes nearly unnoticed if we talk about 'squaring a matrix', though, which is the case of f(x) = x^2 and M an n\times ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
