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Ultrabarrelled Space
In functional analysis and related areas of mathematics, an ultrabarrelled space is a topological vector spaces (TVS) for which every ultrabarrel is a neighbourhood of the origin. Definition A subset B_0 of a TVS X is called an ultrabarrel if it is a closed and balanced subset of X and if there exists a sequence \left(B_i\right)_^ of closed balanced and absorbing subsets of X such that B_ + B_ \subseteq B_i for all i = 0, 1, \ldots. In this case, \left(B_i\right)_^ is called a defining sequence for B_0. A TVS X is called ultrabarrelled if every ultrabarrel in X is a neighbourhood of the origin. Properties A locally convex ultrabarrelled space is a barrelled space. Every ultrabarrelled space is a quasi-ultrabarrelled space. Examples and sufficient conditions Complete and metrizable TVSs are ultrabarrelled. If X is a complete locally bounded non-locally convex TVS and if B_0 is a closed balanced and bounded neighborhood of the origin, then B_0 is an ultrabarrel that ... [...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] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Vector Spaces
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 top ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neighbourhood (topology)
In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. It is closely related to the concepts of open set and interior. Intuitively speaking, a neighbourhood of a point is a set of points containing that point where one can move some amount in any direction away from that point without leaving the set. Definitions Neighbourhood of a point If X is a topological space and p is a point in X, then a of p is a subset V of X that includes an open set U containing p, p \in U \subseteq V \subseteq X. This is also equivalent to the point p \in X belonging to the topological interior of V in X. The neighbourhood V need be an open subset X, but when V is open in X then it is called an . Some authors have been known to require neighbourhoods to be open, so it is important to note conventions. A set that is a neighbourhood of each of its points is open since it can be expressed as the union of open sets ... [...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] |
Absorbing Set
In functional analysis and related areas of mathematics an absorbing set in a vector space is a Set (mathematics), set S which can be "inflated" or "scaled up" to eventually always include any given point of the vector space. Alternative terms are Radial set, radial or absorbent set. Every Neighbourhood (mathematics), neighborhood of the origin in every topological vector space is an absorbing subset. Definition Suppose that X is a vector space over the Field (mathematics), field \mathbb of real numbers \R or complex numbers \Complex. Notation Products of scalars and vectors For any -\infty \leq r \leq R \leq \infty, vector x, and subset A \subseteq X, let B_r = \ \quad \text \quad B_ = \ denote the ''open ball'' (respectively, the ''closed ball'') of radius r in \mathbb centered at 0, and let (r, R) x = \ \quad \text \quad (r, R) A = \. Similarly, if K \subseteq \mathbb and k is a scalar then let K A = \, K x = \, k A = \, and \mathbb x = \ = \operatorname \. Balanced co ... [...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 topologies on vect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barrelled Space
In functional analysis and related areas of mathematics, a barrelled space (also written barreled space) is a topological vector space (TVS) for which every barrelled set in the space is a neighbourhood for the zero vector. A barrelled set or a barrel in a topological vector space is a set that is convex, balanced, absorbing, and closed. Barrelled spaces are studied because a form of the Banach–Steinhaus theorem still holds for them. Barrelled spaces were introduced by . Barrels A convex and balanced subset of a real or complex vector space is called a and it is said to be , , or . A or a in a topological vector space (TVS) is a subset that is a closed absorbing disk; that is, a barrel is a convex, balanced, closed, and absorbing subset. Every barrel must contain the origin. If \dim X \geq 2 and if S is any subset of X, then S is a convex, balanced, and absorbing set of X if and only if this is all true of S \cap Y in Y for every 2-dimensional vector subspace Y; thus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-ultrabarrelled Space
In functional analysis and related areas of mathematics, a quasi-ultrabarrelled space is a topological vector spaces (TVS) for which every bornivorous ultrabarrel is a neighbourhood of the origin. Definition A subset ''B''0 of a TVS ''X'' is called a bornivorous ultrabarrel if it is a closed, balanced, and bornivorous subset of ''X'' and if there exists a sequence \left( B_ \right)_^ of closed balanced and bornivorous subsets of ''X'' such that ''B''''i''+1 + ''B''''i''+1 ⊆ ''B''''i'' for all ''i'' = 0, 1, .... In this case, \left( B_ \right)_^ is called a defining sequence for ''B''0. A TVS ''X'' is called quasi-ultrabarrelled if every bornivorous ultrabarrel in ''X'' is a neighbourhood of the origin. Properties A locally convex quasi-ultrabarrelled space is quasi-barrelled. Examples and sufficient conditions Ultrabarrelled spaces and ultrabornological spaces are quasi-ultrabarrelled. Complete and metrizable TVSs are quasi-ultrabarrelled. See also * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Bounded
In mathematics, a function is locally bounded if it is bounded around every point. A family of functions is locally bounded if for any point in their domain all the functions are bounded around that point and by the same number. Locally bounded function A real-valued or complex-valued function f defined on some topological space X is called a if for any x_0 \in X there exists a neighborhood A of x_0 such that f(A) is a bounded set. That is, for some number M > 0 one has , f(x), \leq M \quad \text x \in A. In other words, for each x one can find a constant, depending on x, which is larger than all the values of the function in the neighborhood of x. Compare this with a bounded function, for which the constant does not depend on x. Obviously, if a function is bounded then it is locally bounded. The converse is not true in general (see below). This definition can be extended to the case when f : X \to Y takes values in some metric space (Y, d). Then the inequality above needs to ... [...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] |
Convex Set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex region is a subset that intersects every line into a single line segment (possibly empty). For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex. The boundary of a convex set is always a convex curve. The intersection of all the convex sets that contain a given subset of Euclidean space is called the convex hull of . It is the smallest convex set containing . A convex function is a real-valued function defined on an interval with the property that its epigraph (the set of points on or above the graph of the function) is a convex set. Convex minimization is a subfield of optimization that studies the problem of minimizing convex functions over convex se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
