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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] |
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] |
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] |
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 function, continuous functions. Such a topology is called a and every topological vector space has a Uniform space, uniform topological structure, allowing a notion of uniform convergence and Complete topological vector space, 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 vec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Order
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The relation itself is called a "partial order." The word ''partial'' in the names "partial order" and "partially ordered set" is used as an indication that not every pair of elements needs to be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset. Partial orders thus generalize total orders, in which every pair is comparable. Informal definition A partial order defines a notion of comparison. Two elements ''x'' and ''y'' may stand in any of four mutually exclusive relationships to each other: either ''x'' ''y'', or ''x'' and ''y'' are ''incompar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordered Vector Space
In mathematics, an ordered vector space or partially ordered vector space is a vector space equipped with a partial order that is compatible with the vector space operations. Definition Given a vector space ''X'' over the real numbers R and a preorder ≤ on the set ''X'', the pair is called a preordered vector space and we say that the preorder ≤ is compatible with the vector space structure of ''X'' and call ≤ a vector preorder on ''X'' if for all ''x'', ''y'', ''z'' in ''X'' and ''λ'' in R with the following two axioms are satisfied # implies # implies . If ≤ is a partial order compatible with the vector space structure of ''X'' then is called an ordered vector space and ≤ is called a vector partial order on ''X''. The two axioms imply that translations and positive homotheties are automorphisms of the order structure and the mapping is an isomorphism to the dual order structure. Ordered vector spaces are ordered groups under their addition op ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Theory
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of systems of linear equations and their generalizations. The theory is connected to that of analytic functions because the spectral properties of an operator are related to analytic functions of the spectral parameter. Mathematical background The name ''spectral theory'' was introduced by David Hilbert in his original formulation of Hilbert space theory, which was cast in terms of quadratic forms in infinitely many variables. The original spectral theorem was therefore conceived as a version of the theorem on principal axes of an ellipsoid, in an infinite-dimensional setting. The later discovery in quantum mechanics that spectral theory could explain features of atomic spectra was therefore ... [...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. Prope ... [...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 then let a S ... [...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. Prope ... [...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] |
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] |
