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DF-space
In the mathematical 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 e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
LM-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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Topological Vector Space
In functional analysis and related areas of mathematics, a complete topological vector space is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point x towards which they all get closer. The notion of "points that get progressively closer" is made rigorous by or , which are generalizations of , while "point x towards which they all get closer" means that this Cauchy net or filter converges to x. The notion of completeness for TVSs uses the theory of uniform spaces as a framework to generalize the notion of completeness for metric spaces. But unlike metric-completeness, TVS-completeness does not depend on any metric and is defined for TVSs, including those that are not metrizable or Hausdorff. Completeness is an extremely important property for a topological vector space to possess. The notions of completeness for normed spaces and metrizable TVSs, which are commonly defined in terms ... [...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 barrel is a neighborhood of the origin. Similarly, 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 A nei ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Tensor Product
In functional analysis, an area of mathematics, the projective tensor product of two locally convex topological vector spaces is a natural topological vector space structure on their tensor product. Namely, given locally convex topological vector spaces X and Y, the projective topology, or π-topology, on X \otimes Y is the strongest topology which makes X \otimes Y a locally convex topological vector space such that the canonical map (x,y) \mapsto x \otimes y (from X\times Y to X \otimes Y) is continuous. When equipped with this topology, X \otimes Y is denoted X \otimes_\pi Y and called the projective tensor product of X and Y. It is a particular instance of a topological tensor product. Definitions Let X and Y be locally convex topological vector spaces. Their projective tensor product X \otimes_\pi Y is the unique locally convex topological vector space with underlying vector space X \otimes Y having the following universal property: :For any locally convex topological vect ... [...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 contin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fréchet Space
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces ( normed vector spaces that are complete with respect to the metric induced by the norm). All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically Banach spaces. A Fréchet space X is defined to be a locally convex metrizable topological vector space (TVS) that is complete as a TVS, meaning that every Cauchy sequence in X converges to some point in X (see footnote for more details).Here "Cauchy" means Cauchy with respect to the canonical uniformity that every TVS possess. That is, a sequence x_ = \left(x_m\right)_^ in a TVS X is Cauchy if and only if for all neighborhoods U of the origin in X, x_m - x_n \in U whenever m and n are sufficiently large. Note that this definiti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fréchet–Urysohn Space
In the field of topology, a Fréchet–Urysohn space is a topological space X with the property that for every subset S \subseteq X the closure of S in X is identical to the ''sequential'' closure of S in X. Fréchet–Urysohn spaces are a special type of sequential space. The property is named after Maurice Fréchet and Pavel Urysohn. Definitions Let (X, \tau) be a topological space. The of S in (X, \tau) is the set: \begin \operatorname S :&= S := \left\ \end where \operatorname_X S or \operatorname_ S may be written if clarity is needed. A topological space (X, \tau) is said to be a if \operatorname_X S = \operatorname_X S for every subset S \subseteq X, where \operatorname_X S denotes the closure of S in (X, \tau). Sequentially open/closed sets Suppose that S \subseteq X is any subset of X. A sequence x_1, x_2, \ldots is if there exists a positive integer N such that x_i \in S for all indices i \geq N. The set S is called if every sequence \left(x_i\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander Grothendieck
Alexander Grothendieck, later Alexandre Grothendieck in French (; ; ; 28 March 1928 – 13 November 2014), was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry. His research extended the scope of the field and added elements of commutative algebra, homological algebra, sheaf theory, and category theory to its foundations, while his so-called Grothendieck's relative point of view, "relative" perspective led to revolutionary advances in many areas of pure mathematics. He is considered by many to be the greatest mathematician of the twentieth century. Grothendieck began his productive and public career as a mathematician in 1949. In 1958, he was appointed a research professor at the Institut des Hautes Études Scientifiques, Institut des hautes études scientifiques (IHÉS) and remained there until 1970, when, driven by personal and political convictions, he left following a dispute over military funding. He receive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sequential Space
In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of countability, and all first-countable spaces (notably metric spaces) are sequential. In any topological space (X, \tau), if a convergent sequence is contained in a closed set C, then the limit of that sequence must be contained in C as well. Sets with this property are known as sequentially closed. Sequential spaces are precisely those topological spaces for which sequentially closed sets are in fact closed. (These definitions can also be rephrased in terms of sequentially open sets; see below.) Said differently, any topology can be described in terms of nets (also known as Moore–Smith sequences), but those sequences may be "too long" (indexed by too large an ordinal) to compress into a sequence. Sequential spaces are those topological spa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
