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Normed Lattice
In mathematics, specifically in order theory and functional analysis, a normed lattice is a topological vector lattice that is also a normed space whose unit ball is a solid set. Normed lattices are important in the theory of topological vector lattice In mathematics, specifically in functional analysis and order theory, a topological vector lattice is a Hausdorff topological vector space (TVS) X that has a partial order \,\leq\, making it into vector lattice that is possesses a neighborhood ba ...s. They are closely related to Banach vector lattices, which are normed vector lattices that are also Banach spaces. Properties Every normed lattice is a locally convex vector lattice. The strong dual of a normed lattice is a Banach lattice with respect to the dual norm and canonical order. If it is also a Banach space then its continuous dual space is equal to its Order dual (functional analysis), order dual. Examples Every Banach lattice is a normed lattice. See also ... [...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] |
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] |
Topological Vector Lattice
In mathematics, specifically in functional analysis and order theory, a topological vector lattice is a Hausdorff topological vector space (TVS) X that has a partial order \,\leq\, making it into vector lattice that is possesses a neighborhood base at the origin consisting of solid sets. Ordered vector lattices have important applications in spectral theory. Definition If X is a vector lattice then by the vector lattice operations we mean the following maps: # the three maps X to itself defined by x \mapsto, x , , x \mapsto x^+, x \mapsto x^, and # the two maps from X \times X into X defined by (x, y) \mapsto \sup_ \ and(x, y) \mapsto \inf_ \. If X is a TVS over the reals and a vector lattice, then X is locally solid if and only if (1) its positive cone is a normal cone, and (2) the vector lattice operations are continuous. If X is a vector lattice and an ordered topological vector space that is a Fréchet space in which the positive cone is a normal cone, then the lattice ... [...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] |
Solid Set
In mathematics, specifically in order theory and functional analysis, a subset S of a vector lattice is said to be solid and is called an ideal if for all s \in S and x \in X, if , x, \leq , s, then x \in S. An ordered vector space whose order is Archimedean is said to be ''Archimedean ordered''. If S\subseteq X then the ideal generated by S is the smallest ideal in X containing S. An ideal generated by a singleton set is called a principal ideal in X. Examples The intersection of an arbitrary collection of ideals in X is again an ideal and furthermore, X is clearly an ideal of itself; thus every subset of X is contained in a unique smallest ideal. In a locally convex vector lattice X, the polar Polar may refer to: Geography Polar may refer to: * Geographical pole, either of two fixed points on the surface of a rotating body or planet, at 90 degrees from the equator, based on the axis around which a body rotates * Polar climate, the c ... of every solid neighborhood ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Vector Lattice
Banach (pronounced in German, in Slavic Languages, and or in English) is a Jewish surname of Ashkenazi origin believed to stem from the translation of the phrase " son of man", combining the Hebrew word ''ben'' ("son of") and Arameic ''nasha'' ("man"). Worth mentioning is how the Sephardic surname ''Banache'' presents a variant with the ''-ache'' alternative ultima, common in other Jewish surnames such as ''Farache'', '' Ayache'', ''Nakache'', ''Harache'' or ''Marrache''. Notable people with this surname include: * Stefan Banach (1892–1945), Polish mathematician * Ed Banach (born 1960), American wrestler * Lou Banach (born 1960), American wrestler * Korneliusz Banach (born 25 January 1994), Polish volleyball player * Łukasz Banach, birth name of Norman Leto (born 1980), Polish artist in the fields of painting, film, and new media *Maurice Banach, German footballer *Orest Banach Orest “Orri” Banach ( ua, Орест (Оррі) Банах) is a retired German-American ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term "Fréchet space." Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces. Definition A Banach space is a complete norme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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] |
Order Dual (functional Analysis)
In mathematics, specifically in order theory and functional analysis, the order dual of an ordered vector space X is the set \operatorname\left(X^*\right) - \operatorname\left(X^*\right) where \operatorname\left(X^*\right) denotes the set of all positive linear functionals on X, where a linear function f on X is called positive if for all x \in X, x \geq 0 implies f(x) \geq 0. The order dual of X is denoted by X^+. Along with the related concept of the order bound dual, this space plays an important role in the theory of ordered topological vector spaces. Canonical ordering An element f of the order dual of X is called positive if x \geq 0 implies \operatorname f(x) \geq 0. The positive elements of the order dual form a cone that induces an ordering on X^+ called the canonical ordering. If X is an ordered vector space whose positive cone C is generating (that is, X = C - C) then the order dual with the canonical ordering is an ordered vector space. The order dual is the span ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
