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Countably Barrelled Space
In functional analysis, a topological vector space (TVS) is said to be countably barrelled if every weakly bounded countable union of equicontinuous subsets of its continuous dual space is again equicontinuous. This property is a generalization of barrelled spaces. Definition A TVS ''X'' with continuous dual space X^ is said to be countably barrelled if B^ \subseteq X^ is a weak-* bounded subset of X^ that is equal to a countable union of equicontinuous subsets of X^, then B^ is itself equicontinuous. A Hausdorff locally convex TVS is countably barrelled if and only if each barrel in ''X'' that is equal to the countable intersection of closed convex balanced neighborhoods of 0 is itself a neighborhood of 0. σ-barrelled space A TVS with continuous dual space X^ is said to be σ-barrelled if every weak-* bounded (countable) sequence in X^ is equicontinuous. Sequentially barrelled space A TVS with continuous dual space X^ is said to be sequentially barrelled if every ... [...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] |
H-space
In mathematics, an H-space is a homotopy-theoretic version of a generalization of the notion of topological group, in which the axioms on associativity and inverses are removed. Definition An H-space consists of a topological space , together with an element of and a continuous map , such that and the maps and are both homotopic to the identity map through maps sending to . This may be thought of as a pointed topological space together with a continuous multiplication for which the basepoint is an identity element up to basepoint-preserving homotopy. One says that a topological space is an H-space if there exists and such that the triple is an H-space as in the above definition. Alternatively, an H-space may be defined without requiring homotopies to fix the basepoint , or by requiring to be an exact identity, without any consideration of homotopy. In the case of a CW complex, all three of these definitions are in fact equivalent. Examples and properties The standard ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Quasi-complete
In functional analysis, a topological vector space (TVS) is said to be quasi-complete or boundedly complete if every closed and bounded subset is complete. This concept is of considerable importance for non- metrizable TVSs. Properties * Every quasi-complete TVS is sequentially complete. * In a quasi-complete locally convex space, the closure of the convex hull of a compact subset is again compact. * In a quasi-complete Hausdorff TVS, every precompact subset is relatively compact. * If is a normed space and is a quasi-complete locally convex TVS then the set of all compact linear maps of into is a closed vector subspace of L_b(X;Y). * Every quasi-complete infrabarrelled space is barreled. * If is a quasi-complete locally convex space then every weakly bounded subset of the continuous dual space is strongly bounded. * A quasi-complete nuclear space then has the Heine–Borel property. Examples and sufficient conditions Every complete TVS is quasi-complete. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Countably Quasi-barrelled Space
In functional analysis, a topological vector space (TVS) is said to be countably quasi-barrelled if every strongly bounded countable union of equicontinuous subsets of its continuous dual space is again equicontinuous. This property is a generalization of quasibarrelled spaces. Definition A TVS ''X'' with continuous dual space X^ is said to be countably quasi-barrelled if B^ \subseteq X^ is a strongly bounded subset of X^ that is equal to a countable union of equicontinuous subsets of X^, then B^ is itself equicontinuous. A Hausdorff locally convex TVS is countably quasi-barrelled if and only if each bornivorous barrel in ''X'' that is equal to the countable intersection of closed convex balanced neighborhoods of 0 is itself a neighborhood of 0. σ-quasi-barrelled space A TVS with continuous dual space X^ is said to be σ-quasi-barrelled if every strongly bounded (countable) sequence in X^ is equicontinuous. Sequentially quasi-barrelled space A TVS with continuo ... [...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] |
DF-space
In the field of functional analysis, DF-spaces, also written (''DF'')-spaces are locally convex topological vector space having a property that is shared by locally convex metrizable topological vector spaces. They play a considerable part in the theory of topological tensor products. DF-spaces were first defined by Alexander Grothendieck and studied in detail by him in . Grothendieck was led to introduce these spaces by the following property of strong duals of metrizable spaces: If X is a metrizable locally convex space and V_1, V_2, \ldots is a sequence of convex 0-neighborhoods in X^_b such that V := \cap_ V_i absorbs every strongly bounded set, then V is a 0-neighborhood in X^_b (where X^_b is the continuous dual space of X endowed with the strong dual topology). Definition A locally convex topological vector space (TVS) X is a DF-space, also written (''DF'')-space, if # X is a countably quasi-barrelled space (i.e. every strongly bounded countable union of equicontinuo ... [...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 normed vec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distinguished Space
In functional analysis and related areas of mathematics, distinguished spaces are topological vector spaces (TVSs) having the property that weak-* bounded subsets of their biduals (that is, the strong dual space of their strong dual space) are contained in the weak-* closure of some bounded subset of the bidual. Definition Suppose that X is a locally convex space and let X^ and X^_b denote the strong dual of X (that is, the continuous dual space of X endowed with the strong dual topology). Let X^ denote the continuous dual space of X^_b and let X^_b denote the strong dual of X^_b. Let X^_ denote X^ endowed with the weak-* topology induced by X^, where this topology is denoted by \sigma\left(X^, X^\right) (that is, the topology of pointwise convergence on X^). We say that a subset W of X^ is \sigma\left(X^, X^\right)-bounded if it is a bounded subset of X^_ and we call the closure of W in the TVS X^_ the \sigma\left(X^, X^\right)-closure of W. If B is a subset of X then the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strong Dual
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] |
Semi-reflexive Space
In the area of mathematics known as functional analysis, a semi-reflexive space is a locally convex topological vector space (TVS) ''X'' such that the canonical evaluation map from ''X'' into its bidual (which is the strong dual of the strong dual of ''X'') is bijective. If this map is also an isomorphism of TVSs then it is called reflexive. Semi-reflexive spaces play an important role in the general theory of locally convex TVSs. Since a normable TVS is semi-reflexive if and only if it is reflexive, the concept of semi-reflexivity is primarily used with TVSs that are not normable. Definition and notation Brief definition Suppose that is a topological vector space (TVS) over the field \mathbb (which is either the real or complex numbers) whose continuous dual space, X^, separates points on (i.e. for any x \in X there exists some x^ \in X^ such that x^(x) \neq 0). Let X^_b and X^_ both denote the strong dual of , which is the vector space X^ of continuous linear func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-barrelled Space
In functional analysis and related areas of mathematics, quasibarrelled spaces are topological vector spaces (TVS) for which every bornivorous barrelled set in the space is a neighbourhood of the origin. Quasibarrelled spaces are studied because they are a weakening of the defining condition of barrelled spaces, for which a form of the Banach–Steinhaus theorem holds. Definition A subset B of a topological vector space (TVS) X is called bornivorous if it absorbs all bounded subsets of X; that is, if for each bounded subset S of X, there exists some scalar r such that S \subseteq r B. A barrelled set or a barrel in a TVS is a set which is convex, balanced, absorbing and closed. A quasibarrelled space is a TVS for which every bornivorous barrelled set in the space is a neighbourhood of the origin. Properties A locally convex Hausdorff quasibarrelled space that is sequentially complete is barrelled. A locally convex Hausdorff quasibarrelled space is a Mackey space, qu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
