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Order Summable
In mathematics, specifically in order theory and functional analysis, a sequence of positive elements \left(x_i\right)_^ in a preordered vector space X (that is, x_i \geq 0 for all i) is called order summable if \sup_ \sum_^n x_i exists in X. For any 1 \leq p \leq \infty, we say that a sequence \left(x_i\right)_^ of positive elements of X is of type \ell^p if there exists some z \in X and some sequence \left(c_i\right)_^ in \ell^p such that 0 \leq x_i \leq c_i z for all i. The notion of order summable sequences is related to the completeness of the order topology In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets. If ''X'' is a totally ordered set, t .... See also * * * * References Bibliography * * {{Ordered topological vector spaces Functional analysis ... [...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] |
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] |
Order Topology (functional Analysis)
In mathematics, specifically in order theory and functional analysis, the order topology of an ordered vector space (X, \leq) is the finest locally convex topological vector space (TVS) topology on X for which every order interval is bounded, where an order interval in X is a set of the form [a, b] := \left\ where a and b belong to X. The order topology is an important topology that is used frequently in the theory of ordered topological vector spaces because the topology stems directly from the algebraic and order theoretic properties of (X, \leq), rather than from some topology that X starts out having. This allows for establishing intimate connections between this topology and the algebraic and order theoretic properties of (X, \leq). For many ordered topological vector spaces that occur in analysis, their topologies are identical to the order topology. Definitions The family of all locally convex topologies on X for which every order interval is bounded is non-empty ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
