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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 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; thus ... [...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] |
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] |
Strong Dual Topology
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-Steinhaus Theorem
In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field. In its basic form, it asserts that for a family of continuous linear operators (and thus bounded operators) whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operator norm. The theorem was first published in 1927 by Stefan Banach and Hugo Steinhaus, but it was also proven independently by Hans Hahn. Theorem The completeness of X enables the following short proof, using the Baire category theorem. There are also simple proofs not using the Baire theorem . Corollaries The above corollary does claim that T_n converges to T in operator norm, that is, uniformly on bounded sets. However, since \left\ is bounded in operator norm, and the limit operator T is continuous, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Graph
In mathematics, particularly in functional analysis and topology, closed graph is a property of functions. A function between topological spaces has a closed graph if its graph is a closed subset of the product space . A related property is open graph. This property is studied because there are many theorems, known as closed graph theorems, giving conditions under which a function with a closed graph is necessarily continuous. One particularly well-known class of closed graph theorems are the closed graph theorems in functional analysis. Definitions Graphs and set-valued functions :Definition and notation: The graph of a function is the set ::. :Notation: If is a set then the power set of , which is the set of all subsets of , is denoted by or . :Definition: If and are sets, a set-valued function in on (also called a -valued multifunction on ) is a function with domain that is valued in . That is, is a function on such that for every , is a subset of ... [...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] |
F-space
In functional analysis, an F-space is a vector space X over the real or complex numbers together with a metric d : X \times X \to \R such that # Scalar multiplication in X is continuous with respect to d and the standard metric on \R or \Complex. # Addition in X is continuous with respect to d. # The metric is translation-invariant; that is, d(x + a, y + a) = d(x, y) for all x, y, a \in X. # The metric space (X, d) is complete. The operation x \mapsto \, x\, := d(0, x) is called an F-norm, although in general an F-norm is not required to be homogeneous. By translation-invariance, the metric is recoverable from the F-norm. Thus, a real or complex F-space is equivalently a real or complex vector space equipped with a complete F-norm. Some authors use the term rather than , but usually the term "Fréchet space" is reserved for locally convex F-spaces. Some other authors use the term "F-space" as a synonym of "Fréchet space", by which they mean a locally convex complete metri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Baire Space
In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior. According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are examples of Baire spaces. The Baire category theorem combined with the properties of Baire spaces has numerous applications in topology, geometry, analysis, in particular functional analysis. Bourbaki introduced the term "Baire space" in honor of René Baire, who investigated the Baire category theorem in the context of Euclidean space \R^n in his 1899 thesis. Definition The definition that follows is based on the notions of meagre (or first category) set (namely, a set that is a countable union of sets whose closure has empty interior) and nonmeagre (or second category) set (namely, a set that is not meagre). See the corresponding article for details. A topological space X is called a Baire space if it satisfies any of the follo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indiscrete Topology
In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the consequence that all points of the space are "lumped together" and cannot be distinguished by topological means. Every indiscrete space is a pseudometric space in which the distance between any two points is zero. Details The trivial topology is the topology with the least possible number of open sets, namely the empty set and the entire space, since the definition of a topology requires these two sets to be open. Despite its simplicity, a space ''X'' with more than one element and the trivial topology lacks a key desirable property: it is not a T0 space. Other properties of an indiscrete space ''X''—many of which are quite unusual—include: * The only closed sets are the empty set and ''X''. * The only possible basis of ''X'' is . ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Dual Space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by constants. The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the . When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called the ''continuous dual space''. Dual vector spaces find application in many branches of mathematics that use vector spaces, such as in tensor analysis with finite-dimensional vector spaces. When applied to vector spaces of functions (which are typically infinite-dimensional), dual spaces are used to describe measures, distributions, and Hilbert spaces. Consequently, the dual space is an important concept in functional analysis. Early terms for ''dual'' include ''polarer Raum'' ah ... [...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] |
Bornivorous Set
In functional analysis, a subset of a real or complex vector space X that has an associated vector bornology \mathcal is called bornivorous and a bornivore if it absorbs every element of \mathcal. If X is a topological vector space (TVS) then a subset S of X is bornivorous if it is bornivorous with respect to the von-Neumann bornology of X. Bornivorous sets play an important role in the definitions of many classes of topological vector spaces, particularly bornological spaces. Definitions If X is a TVS then a subset S of X is called and a if S absorbs every bounded subset of X. An absorbing disk in a locally convex space is bornivorous if and only if its Minkowski functional is locally bounded (i.e. maps bounded sets to bounded sets). Infrabornivorous sets and infrabounded maps A linear map between two TVSs is called if it maps Banach disks to bounded disks. A disk in X is called if it absorbs every Banach disk. An absorbing disk in a locally conv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
