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Normal Cone (functional Analysis)
In mathematics, specifically in order theory and functional analysis, if C is a cone at the origin in a topological vector space X such that 0 \in C and if \mathcal is the neighborhood filter at the origin, then C is called normal if \mathcal = \left \mathcal \rightC, where \left \mathcal \rightC := \left\ and where for any subset S \subseteq X, C := (S + C) \cap (S - C) is the C-saturatation of S. Normal cones play an important role in the theory of ordered topological vector spaces and topological vector lattices. Characterizations If C is a cone in a TVS X then for any subset S \subseteq X let C := \left(S + C\right) \cap \left(S - C\right) be the C-saturated hull of S \subseteq X and for any collection \mathcal of subsets of X let \left \mathcal \rightC := \left\. If C is a cone in a TVS X then C is normal if \mathcal = \left \mathcal \rightC, where \mathcal is the neighborhood filter at the origin. If \mathcal is a collection of subsets of X and if \mathcal is a ... [...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 differe ... [...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, norm, topology, etc.) and the 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 spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations. The usage of the word '' functional'' as a noun goes back to the calculus of variations, implying a function whose argument is a function. The term was first used in Hadamard's 1910 book on that subject. However, the general concept of a functional had previously been introduced in 1887 by the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Vector Space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is also a topological space with the property that the vector space operations (vector addition and scalar multiplication) are also continuous functions. Such a topology is called a and every topological vector space has a uniform topological structure, allowing a notion of uniform convergence and completeness. Some authors also require that the space is a Hausdorff space (although this article does not). One of the most widely studied categories of TVSs are locally convex topological vector spaces. This article focuses on TVSs that are not necessarily locally convex. Banach spaces, Hilbert spaces and Sobolev spaces are other well-known examples of TVSs. Many topological vector spaces are spaces of functions, or linear operators acting on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neighborhood Filter
In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x. Definitions Neighbourhood of a point or set An of a point (or subset) x in a topological space X is any Open set, open subset U of X that contains x. A is any subset N \subseteq X that contains open neighbourhood of x; explicitly, N is a neighbourhood of x in X if and only if there exists some open subset U with x \in U \subseteq N. Equivalently, a neighborhood of x is any set that contains x in its interior (topology), topological interior. Importantly, a "neighbourhood" does have to be an open set; those neighbourhoods that also happen to be open sets are known as "open neighbourhoods." Similarly, a neighbourhood that is also a Closed set, closed (respectively, compact space, compact, connected space, connected, etc.) set is called a (respectively, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cone-saturated
In mathematics, specifically in order theory and functional analysis, if C is a cone at 0 in a vector space X such that 0 \in C, then a subset S \subseteq X is said to be C-saturated if S = C, where C := (S + C) \cap (S - C). Given a subset S \subseteq X, the C-saturated hull of S is the smallest C-saturated subset of X that contains S. If \mathcal is a collection of subsets of X then \left \mathcal \rightC := \left\. If \mathcal is a collection of subsets of X and if \mathcal is a subset of \mathcal then \mathcal is a fundamental subfamily of \mathcal if every T \in \mathcal is contained as a subset of some element of \mathcal. If \mathcal is a family of subsets of a TVS X then a cone C in X is called a \mathcal-cone if \left\ is a fundamental subfamily of \mathcal and C is a strict \mathcal-cone if \left\ is a fundamental subfamily of \mathcal. C-saturated sets play an important role in the theory of ordered topological vector spaces and topological vector lattices. P ... [...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 ... [...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 th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Balanced Set
In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field \mathbb with an absolute value function , \cdot , ) is a set S such that a S \subseteq S for all scalars a satisfying , a, \leq 1. The balanced hull or balanced envelope of a set S is the smallest balanced set containing S. The balanced core of a subset S is the largest balanced set contained in S. Balanced sets are ubiquitous in functional analysis because every neighborhood of the origin in every topological vector space (TVS) contains a balanced neighborhood of the origin and every convex neighborhood of the origin contains a balanced convex neighborhood of the origin (even if the TVS is not locally convex). This neighborhood can also be chosen to be an open set or, alternatively, a closed set. Definition Let X be a vector space over the field \mathbb of real or complex numbers. Notation If S is a set, a is a scalar, and B \subseteq \mathbb t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cone-saturated
In mathematics, specifically in order theory and functional analysis, if C is a cone at 0 in a vector space X such that 0 \in C, then a subset S \subseteq X is said to be C-saturated if S = C, where C := (S + C) \cap (S - C). Given a subset S \subseteq X, the C-saturated hull of S is the smallest C-saturated subset of X that contains S. If \mathcal is a collection of subsets of X then \left \mathcal \rightC := \left\. If \mathcal is a collection of subsets of X and if \mathcal is a subset of \mathcal then \mathcal is a fundamental subfamily of \mathcal if every T \in \mathcal is contained as a subset of some element of \mathcal. If \mathcal is a family of subsets of a TVS X then a cone C in X is called a \mathcal-cone if \left\ is a fundamental subfamily of \mathcal and C is a strict \mathcal-cone if \left\ is a fundamental subfamily of \mathcal. C-saturated sets play an important role in the theory of ordered topological vector spaces and topological vector lattices. P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infrabarreled
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 A barrel or cask is a hollow cylindrical container with a bulging center, longer than it is wide. They are traditionally made of wooden staves and bound by wooden or metal hoops. The word vat is often used for large containers for liquids, ... 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 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. Ev ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Compact
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which every point has a compact neighborhood. In mathematical analysis locally compact spaces that are Hausdorff are of particular interest; they are abbreviated as LCH spaces. Formal definition Let ''X'' be a topological space. Most commonly ''X'' is called locally compact if every point ''x'' of ''X'' has a compact neighbourhood, i.e., there exists an open set ''U'' and a compact set ''K'', such that x\in U\subseteq K. There are other common definitions: They are all equivalent if ''X'' is a Hausdorff space (or preregular). But they are not equivalent in general: :1. every point of ''X'' has a compact neighbourhood. :2. every point of ''X'' has a closed compact neighbourhood. :2′. every point of ''X'' has a relatively compact neighbou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Convex
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 topologie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
