|
Quasi-ultrabarrelled Space
In functional analysis and related areas of mathematics, a quasi-ultrabarrelled space is a topological vector spaces (TVS) for which every bornivorous ultrabarrel is a neighbourhood of the origin. Definition A subset ''B''0 of a TVS ''X'' is called a bornivorous ultrabarrel if it is a closed, balanced, and bornivorous subset of ''X'' and if there exists a sequence \left( B_ \right)_^ of closed balanced and bornivorous subsets of ''X'' such that ''B''''i''+1 + ''B''''i''+1 ⊆ ''B''''i'' for all ''i'' = 0, 1, .... In this case, \left( B_ \right)_^ is called a defining sequence for ''B''0. A TVS ''X'' is called quasi-ultrabarrelled if every bornivorous ultrabarrel in ''X'' is a neighbourhood of the origin. Properties A locally convex quasi-ultrabarrelled space is quasi-barrelled. Examples and sufficient conditions Ultrabarrelled spaces and ultrabornological spaces are quasi-ultrabarrelled. Complete and metrizable TVSs are quasi-ultrabarrelled. See also * ... [...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] |
Ultrabornological Space
In functional analysis, a topological vector space (TVS) X is called ultrabornological if every bounded linear operator from X into another TVS is necessarily continuous. A general version of the closed graph theorem holds for ultrabornological spaces. Ultrabornological spaces were introduced by Alexander Grothendieck (Grothendieck 955, p. 17"espace du type (β)"). Definitions Let X be a topological vector space (TVS). Preliminaries A disk is a convex and balanced set. A disk in a TVS X is called bornivorous if it absorbs every bounded subset of X. A linear map between two TVSs is called infrabounded if it maps Banach disks to bounded disks. A disk D in a TVS X is called infrabornivorous if it satisfies any of the following equivalent conditions: D absorbs every Banach disks in X. while if X locally convex then we may add to this list: the gauge of D is an infrabounded map; while if X locally convex and Hausdorff then we may add to this list: D absorbs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annales De L'Institut Fourier
The ''Annales de l'Institut Fourier'' is a French mathematical journal publishing papers in all fields of mathematics. It was established in 1949. The journal publishes one volume per year, consisting of six issues. The current editor-in-chief is Hervé Pajot. Articles are published either in English or in French. The journal is indexed in ''Mathematical Reviews'', ''Zentralblatt MATH'' and the Web of Science. According to the ''Journal Citation Reports'', the journal had a 2008 impact factor of 0.804. 2008 Journal Citation Reports, Science Edition, Thomson Scientific Thomson Scientific was one of the six (later five) strategic business units of The Thomson Corporation, beginning in 2007, after being separated from Thomson Scientific & Healthcare. Following the merger of Thomson with Reuters Group to form Thom ..., 2008. References External links * Mathematics journals Academic journals established in 1949 Multilingual journals Bimonthly journals Open access journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Boundedness Principle
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 ... [...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] |
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] |
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] |
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] |
Ultrabarrelled Space
In functional analysis and related areas of mathematics, an ultrabarrelled space is a topological vector spaces (TVS) for which every ultrabarrel is a neighbourhood of the origin. Definition A subset B_0 of a TVS X is called an ultrabarrel if it is a closed and balanced subset of X and if there exists a sequence \left(B_i\right)_^ of closed balanced and absorbing subsets of X such that B_ + B_ \subseteq B_i for all i = 0, 1, \ldots. In this case, \left(B_i\right)_^ is called a defining sequence for B_0. A TVS X is called ultrabarrelled if every ultrabarrel in X is a neighbourhood of the origin. Properties A locally convex ultrabarrelled space is a barrelled space. Every ultrabarrelled space is a quasi-ultrabarrelled space. Examples and sufficient conditions Complete and metrizable TVSs are ultrabarrelled. If X is a complete locally bounded non-locally convex TVS and if B_0 is a closed balanced and bounded neighborhood of the origin, then B_0 is an ultrabarrel that ... [...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] |
Quasibarrelled 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 (topology), 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 (mathematics), set which is Convex set, convex, Balanced set, balanced, Absorbing set, absorbing and Closed set, closed. A quasibarrelled space is a TVS for which every bornivorous barrelled set in the space is a Neighbourhood (topology), neighbourhood of the origin. Properties A locally convex Hausdorff quasibarrelled sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
