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Absorbent Set
In functional analysis and related areas of mathematics an absorbing set in a vector space is a set S which can be "inflated" or "scaled up" to eventually always include any given point of the vector space. Alternative terms are radial or absorbent set. Every neighborhood of the origin in every topological vector space is an absorbing subset. Definition Notation for scalars Suppose that X is a vector space over the field \mathbb of real numbers \R or complex numbers \Complex, and for any -\infty \leq r \leq \infty, let B_r = \ \quad \text \quad B_ = \ denote the ''open ball'' (respectively, the ''closed ball'') of radius r in \mathbb centered at 0. Define the product of a set K \subseteq \mathbb of scalars with a set A of vectors as K A = \, and define the product of K \subseteq \mathbb with a single vector x as K x = \. Preliminaries Balanced core and balanced hull A subset S of X is said to be '' '' if a s \in S for all s \in S and all scalars a satisfying , a, \leq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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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 (for example, Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, or Topological space#Definitions, topology) and the linear transformation, linear functions defined on these spaces and suitably respecting these structures. 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, for example, continuous function, continuous or unitary operator, unitary 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 v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Balanced Hull
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 set 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] [Amazon] |
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Subspace Topology
In topology and related areas of mathematics, a subspace of a topological space (''X'', ''𝜏'') is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''𝜏'' called the subspace topology (or the relative topology, or the induced topology, or the trace topology).; see Section 26.2.4. Submanifolds, p. 59 Definition Given a topological space (X, \tau) and a subset S of X, the subspace topology on S is defined by :\tau_S = \lbrace S \cap U \mid U \in \tau \rbrace. That is, a subset of S is open in the subspace topology if and only if it is the intersection of S with an open set in (X, \tau). If S is equipped with the subspace topology then it is a topological space in its own right, and is called a subspace of (X, \tau). Subsets of topological spaces are usually assumed to be equipped with the subspace topology unless otherwise stated. Alternatively we can define the subspace topology for a subset S of X as the coarsest topology for which the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Hausdorff Vector Topology On A 1-dimensional Vector Space
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Hausdorff may refer to: People * Felix Hausdorff (1868–1942), German mathematician after whom Hausdorff spaces are named *Natasha Hausdorff (born 1989), British barrister, international news commentator, and Israel advocate Other * A Hausdorff space, when used as an adjective, as in "the real line is Hausdorff" * Hausdorff dimension, a measure theoretic concept of dimension * Hausdorff distance or Hausdorff metric, which measures how far two compact non-empty subsets of a metric space are from each other * Hausdorff density * Hausdorff maximal principle * Hausdorff measure * Hausdorff moment problem * Hausdorff paradox The Hausdorff paradox is a paradox in mathematics named after Felix Hausdorff. It involves the sphere (the surface of a 3-dimensional ball in ). It states that if a certain countable subset is removed from , then the remainder can be divided into ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Euclidean Metric
In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, and therefore is occasionally called the Pythagorean distance. These names come from the ancient Greek mathematicians Euclid and Pythagoras. In the Greek deductive geometry exemplified by Euclid's ''Elements'', distances were not represented as numbers but line segments of the same length, which were considered "equal". The notion of distance is inherent in the compass tool used to draw a circle, whose points all have the same distance from a common center point. The connection from the Pythagorean theorem to distance calculation was not made until the 18th century. The distance between two objects that are not points is usually defined to be the smallest distance among pairs of points from the two objects. Formulas are known for computing distances betw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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TVS-isomorphism
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. Other well-known examples of TVSs include Banach spaces, Hilbert spaces and Sobolev spaces. Many topological vector spaces are spaces of functions, or linear operators act ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Trivial Topology
In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the consequence that all points of the space are "lumped together" and cannot be distinguished by topological means. Every indiscrete space can be viewed as a pseudometric space in which the distance between any two points is zero. Details The trivial topology is the topology with the least possible number of open sets, namely the empty set and the entire space, since the definition of a topology requires these two sets to be open. Despite its simplicity, a space ''X'' with more than one element and the trivial topology lacks a key desirable property: it is not a T0 space. Other properties of an indiscrete space ''X''—many of which are quite unusual—include: * The only closed sets are the empty set and ''X''. * The only possible basi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Absorb A Set
Absorption may refer to: Chemistry and biology *Absorption (biology), digestion **Absorption (small intestine) *Absorption (chemistry), diffusion of particles of gas or liquid into liquid or solid materials *Absorption (skin), a route by which substances enter the body through the skin *Absorption (pharmacology), absorption of drugs into the body Physics and chemical engineering *Absorption (acoustics), absorption of sound waves by a material *Absorption (electromagnetic radiation), absorption of light or other electromagnetic radiation by a material *Absorption air conditioning, a type of solar air conditioning *Absorption refrigerator, a refrigerator that runs on surplus heat rather than electricity *Dielectric absorption, the inability of a charged capacitor to completely discharge when briefly discharged Mathematics and economics *Absorption (economics), the total demand of an economy for goods and services both from within and without *Absorption (logic), one of the rules of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Unit Circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. In topology, it is often denoted as because it is a one-dimensional unit -sphere. If is a point on the unit circle's circumference, then and are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, and satisfy the equation x^2 + y^2 = 1. Since for all , and since the reflection of any point on the unit circle about the - or -axis is also on the unit circle, the above equation holds for all points on the unit circle, not only those in the first quadrant. The interior of the unit circle is called the open unit disk, while the interior of the unit circle combined with the unit circle itself is called the closed unit disk. One may also use other notions of "dis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Bornivorous Set
In functional analysis, a subset of a real or complex vector space X that has an associated vector bornology \mathcal is called bornivorous and a bornivore if it absorbs every element of \mathcal. If X is a topological vector space (TVS) then a subset S of X is bornivorous if it is bornivorous with respect to the von-Neumann bornology of X. Bornivorous sets play an important role in the definitions of many classes of topological vector spaces, particularly bornological spaces. Definitions If X is a TVS then a subset S of X is called and a if S absorbs every bounded subset of X. An absorbing disk in a locally convex space is bornivorous if and only if its Minkowski functional is locally bounded (i.e. maps bounded sets to bounded sets). Infrabornivorous sets and infrabounded maps A linear map between two TVSs is called if it maps Banach disks to bounded disks. A disk in X is called if it absorbs every Banach disk. An absorbing disk in a locally conve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |