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Locally Convex Vector Lattice
In mathematics, specifically in order theory and functional analysis, a locally convex vector lattice (LCVL) is a topological vector lattice that is also a locally convex space. LCVLs are important in the theory of topological vector lattices. Lattice semi-norms The Minkowski functional of a convex, absorbing, and solid set is a called a lattice semi-norm. Equivalently, it is a semi-norm p such that , y, \leq , x, implies p(y) \leq p(x). The topology of a locally convex vector lattice is generated by the family of all continuous lattice semi-norms. Properties Every locally convex vector lattice possesses a neighborhood base at the origin consisting of convex balanced solid absorbing sets. The strong dual of a locally convex vector lattice X is an order complete locally convex vector lattice (under its canonical order) and it is a solid subspace of the order dual of X; moreover, if X is a barreled space then the continuous dual space of X is a band in the order dual ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Theory
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and provides basic definitions. A list of order-theoretic terms can be found in the order theory glossary. Background and motivation Orders are everywhere in mathematics and related fields like computer science. The first order often discussed in primary school is the standard order on the natural numbers e.g. "2 is less than 3", "10 is greater than 5", or "Does Tom have fewer cookies than Sally?". This intuitive concept can be extended to orders on other sets of numbers, such as the integers and the reals. The idea of being greater than or less than another number is one of the basic intuitions of number systems (compare with numeral systems) in general (although one usually is also interested in the actual difference ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reflexive Space
In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space (TVS) for which the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is an isomorphism of TVSs. Since a normable TVS is reflexive if and only if it is semi-reflexive, every normed space (and so in particular, every Banach space) X is reflexive if and only if the canonical evaluation map from X into its bidual is surjective; in this case the normed space is necessarily also a Banach space. In 1951, R. C. James discovered a Banach space, now known as James' space, that is reflexive but is nevertheless isometrically isomorphic to its bidual (any such isomorphism is thus necessarily the canonical evaluation map). Reflexive spaces play an important role in the general theory of locally convex TVSs and in the theory of Banach spaces in particular. Hilbert spaces are prominent examples of reflexive Banach spaces. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Lattice
In the mathematical disciplines of in functional analysis and order theory, a Banach lattice is a complete normed vector space with a lattice order, such that for all , the implication \Rightarrow holds, where the absolute value is defined as , x, = x \vee -x\text Examples and constructions Banach lattices are extremely common in functional analysis, and "every known example n 1948of a Banach space asalso a vector lattice." In particular: * , together with its absolute value as a norm, is a Banach lattice. * Let be a topological space, a Banach lattice and the space of continuous bounded functions from to with norm \, f\, _ = \sup_ \, f(x)\, _Y\text Then is a Banach lattice under the pointwise partial order: \Leftrightarrow(\forall x\in X)(f(x)\leq g(x))\text Examples of non-lattice Banach spaces are now known; James' space is one such.Kania, Tomasz (12 April 2017).Answerto "Banach space that is not a Banach lattice" (accessed 13 August 2022). ''Mathematics St ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sequentially Complete
In mathematics, specifically in topology and functional analysis, a subspace of a uniform space is said to be sequentially complete or semi-complete if every Cauchy sequence in converges to an element in . is called sequentially complete if it is a sequentially complete subset of itself. Sequentially complete topological vector spaces Every topological vector space is a uniform space so the notion of sequential completeness can be applied to them. Properties of sequentially complete topological vector spaces #A bounded sequentially complete disk in a Hausdorff topological vector space is a Banach disk. #A Hausdorff locally convex space that is sequentially complete and bornological is ultrabornological. Examples and sufficient conditions #Every complete space is sequentially complete but not conversely. #A metrizable space then it is complete if and only if it is sequentially complete. #Every complete topological vector space is quasi-complete and every quasi-c ... [...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] |
Majorized
In mathematics, majorization is a preorder on vectors of real numbers. Let ^_,\ i=1,\,\ldots,\,n denote the i-th largest element of the vector \mathbf\in\mathbb^n. Given \mathbf,\ \mathbf \in \mathbb^n, we say that \mathbf weakly majorizes (or dominates) \mathbf from below (or equivalently, we say that \mathbf is weakly majorized (or dominated) by \mathbf from below) denoted as \mathbf \succ_w \mathbf if \sum_^k x_^ \geq \sum_^k y_^ for all k=1,\,\dots,\,d. If in addition \sum_^d x_i^ = \sum_^d y_i^, we say that \mathbf majorizes (or dominates) \mathbf , written as \mathbf \succ \mathbf , or equivalently, we say that \mathbf is majorized (or dominated) by \mathbf. The order of the entries of the vectors \mathbf or \mathbf does not affect the majorization, e.g., the statement (1,2)\prec (0,3) is simply equivalent to (2,1)\prec (3,0). As a consequence, majorization is not a partial order, since \mathbf \succ \mathbf and \mathbf \succ \mathbf do not imply \mathbf = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Order (mathematics)
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Terms such as regular order, regular orders, and the like can refer to: * Regular order (United States Congress), a process of governance * Normal order (other) * Regular clergy, members of a religious order who live according to a prescribed rule See also * Order (other) * Regular (other) The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instrumen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Convergent
In mathematics, specifically in order theory and functional analysis, a filter \mathcal in an order complete vector lattice X is order convergent if it contains an order bounded subset (that is, is contained in an interval of the form , b:= \) and if \mathcal, \sup \left\ = \inf \left\, where \operatorname(X) is the set of all order bounded subsets of ''X'', in which case this common value is called the order limit of \mathcal in X. Order convergence plays an important role in the theory of vector lattices because the definition of order convergence does not depend on any topology. Definition A net \left(x_\right)_ in a vector lattice X is said to decrease to x_0 \in X if \alpha \leq \beta implies x_ \leq x_ and x_0 = inf \left\ in X. A net \left(x_\right)_ in a vector lattice X is said to order-converge to x_0 \in X if there is a net \left(y_\right)_ in X that decreases to 0 and satisfies \left, x_ - x_0\ \leq y_ for all \alpha \in A. Order continuity A linear map T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-interior Point
In mathematics, specifically in order theory and functional analysis, an element x of an ordered topological vector space X is called a quasi-interior point of the positive cone C of X if x \geq 0 and if the order interval , x:= \ is a total subset of X; that is, if the linear span of , x/math> is a dense subset of X. Properties If X is a separable metrizable locally convex ordered topological vector space In mathematics, specifically in functional analysis and order theory, an ordered topological vector space, also called an ordered TVS, is a topological vector space (TVS) ''X'' that has a partial order ≤ making it into an ordered vector space whos ... whose positive cone C is a complete and total subset of X, then the set of quasi-interior points of C is dense in C. Examples If 1 \leq p \, 0 almost everywhere (with respect to \mu). A point in L^\infty(\mu) is quasi-interior to the positive cone C if and only if it is interior to C. See also * * Referen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordered Topological Vector Space
In mathematics, specifically in functional analysis and order theory, an ordered topological vector space, also called an ordered TVS, is a topological vector space (TVS) ''X'' that has a partial order ≤ making it into an ordered vector space whose positive cone C := \left\ is a closed subset of ''X''. Ordered TVS have important applications in spectral theory. Normal cone If ''C'' is a cone in a TVS ''X'' then ''C'' is normal if \mathcal = \left \mathcal \right, where \mathcal is the neighborhood filter at the origin, \left \mathcal \right = \left\, and := \left(U + C\right) \cap \left(U - C\right) is the ''C''-saturated hull of a subset ''U'' of ''X''. If ''C'' is a cone in a TVS ''X'' (over the real or complex numbers), then the following are equivalent: # ''C'' is a normal cone. # For every filter \mathcal in ''X'', if \lim \mathcal = 0 then \lim \left \mathcal \right = 0. # There exists a neighborhood base \mathcal in ''X'' such that B \in \mathcal implies \left B \cap ... [...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 topolo ... [...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] |
