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List Of Vector Spaces In Mathematics
This is a list of vector spaces in abstract mathematics, by Wikipedia page. *Banach space * Besov space *Bochner space *Dual space *Euclidean space *Fock space *Fréchet space *Hardy space *Hilbert space * Hölder space *LF-space * L''p'' space *Minkowski space *Montel space *Morrey–Campanato space *Orlicz space *Riesz space *Schwartz space *Sobolev space *Tsirelson space In mathematics, especially in functional analysis, the Tsirelson space is the first example of a Banach space in which neither an ℓ ''p'' space nor a ''c''0 space can be embedded. The Tsirelson space is reflexive. It was introduced by B. ... {{DEFAULTSORT:Vector spaces in mathematics * Linear algebra Mathematics-related lists ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space. Vector spaces generalize Euclidean vectors, which allow modeling of physical quantities, such as forces and velocity, that have not only a magnitude, but also a direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying systems of li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lp Space
In mathematics, the spaces are function spaces defined using a natural generalization of the -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Bourbaki group they were first introduced by Frigyes Riesz . spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Because of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used also in the theoretical discussion of problems in physics, statistics, economics, finance, engineering, and other disciplines. Applications Statistics In statistics, measures of central tendency and statistical dispersion, such as the mean, median, and standard deviation, are defined in terms of metrics, and measures of central tendency can be characterized as solutions to variational problems. In penalized regression, "L1 penalty" and "L2 penalty" refer ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tsirelson Space
In mathematics, especially in functional analysis, the Tsirelson space is the first example of a Banach space in which neither an ℓ ''p'' space nor a ''c''0 space can be embedded. The Tsirelson space is reflexive. It was introduced by B. S. Tsirelson in 1974. The same year, Figiel and Johnson published a related article () where they used the notation ''T'' for the ''dual'' of Tsirelson's example. Today, the letter ''T'' is the standard notationsee for example , p. 8; , p. 95; ''The Handbook of the Geometry of Banach Spaces'', vol. 1, p. 276; vol. 2, p. 1060, 1649. for the dual of the original example, while the original Tsirelson example is denoted by ''T''*. In ''T''* or in ''T'', no subspace is isomorphic, as Banach space, to an ''ℓ'' ''p'' space, 1 ≤ ''p'' < ∞, or to ''c''0. All classical Banach spaces known to , spaces of |
Sobolev Space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, i.e. a Banach space. Intuitively, a Sobolev space is a space of functions possessing sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function. Sobolev spaces are named after the Russian mathematician Sergei Sobolev. Their importance comes from the fact that weak solutions of some important partial differential equations exist in appropriate Sobolev spaces, even when there are no strong solutions in spaces of continuous functions with the derivatives understood in the classical sense. Motivation In this section and throughout the article \Omega is an open subset of \R^n. There are ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schwartz Space
In mathematics, Schwartz space \mathcal is the function space of all functions whose derivatives are rapidly decreasing. This space has the important property that the Fourier transform is an automorphism on this space. This property enables one, by duality, to define the Fourier transform for elements in the dual space \mathcal^* of \mathcal, that is, for tempered distributions. A function in the Schwartz space is sometimes called a Schwartz function. Schwartz space is named after French mathematician Laurent Schwartz. Definition Let \mathbb be the set of non-negative integers, and for any n \in \mathbb, let \mathbb^n := \underbrace_ be the ''n''-fold Cartesian product. The ''Schwartz space'' or space of rapidly decreasing functions on \mathbb^n is the function spaceS \left(\mathbb^n, \mathbb\right) := \left \,where C^(\mathbb^n, \mathbb) is the function space of smooth functions from \mathbb^n into \mathbb, and\, f\, _:= \sup_ \left, x^\alpha (D^ f)(x) \.Here, \sup de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riesz Space
In mathematics, a Riesz space, lattice-ordered vector space or vector lattice is a partially ordered vector space where the order structure is a lattice. Riesz spaces are named after Frigyes Riesz who first defined them in his 1928 paper ''Sur la décomposition des opérations fonctionelles linéaires''. Riesz spaces have wide-ranging applications. They are important in measure theory, in that important results are special cases of results for Riesz spaces. For example, the Radon–Nikodym theorem follows as a special case of the Freudenthal spectral theorem. Riesz spaces have also seen application in mathematical economics through the work of Greek-American economist and mathematician Charalambos D. Aliprantis. Definition Preliminaries If X is an ordered vector space (which by definition is a vector space over the reals) and if S is a subset of X then an element b \in X is an upper bound (resp. lower bound) of S if s \leq b (resp. s \geq b) for all s \in S. An elemen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orlicz Space
In mathematical analysis, and especially in real, harmonic analysis and functional analysis, an Orlicz space is a type of function space which generalizes the ''L''''p'' spaces. Like the ''L''''p'' spaces, they are Banach spaces. The spaces are named for Władysław Orlicz, who was the first to define them in 1932. Besides the ''L''''p'' spaces, a variety of function spaces arising naturally in analysis are Orlicz spaces. One such space ''L'' log+ ''L'', which arises in the study of Hardy–Littlewood maximal functions, consists of measurable functions ''f'' such that the integral :\int_ , f(x), \log^+ , f(x), \,dx < \infty. Here log+ is the positive part of the logarithm. Also included in the class of Orlicz spaces are many of the most important |
Morrey–Campanato Space
In mathematics, the Morrey–Campanato spaces (named after Charles B. Morrey, Jr. and Sergio Campanato) L^(\Omega) are Banach spaces which extend the notion of functions of bounded mean oscillation, describing situations where the oscillation of the function in a ball is proportional to some power of the radius other than the dimension. They are used in the theory of elliptic partial differential equations, since for certain values of \lambda, elements of the space L^(\Omega) are Hölder continuous functions over the domain \Omega. The seminorm of the Morrey spaces is given by :\bigl( \bigr)^p = \sup_ \frac \int_ , u(y) , ^p dy. When \lambda = 0, the Morrey space is the same as the usual L^p space. When \lambda = n, the spatial dimension, the Morrey space is equivalent to L^\infty, due to the Lebesgue differentiation theorem. When \lambda > n, the space contains only the 0 function. Note that this is a norm for p \geq 1 . The seminorm of the Campanato space is given by :\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Montel Space
In functional analysis and related areas of mathematics, a Montel space, named after Paul Montel, is any topological vector space (TVS) in which an analog of Montel's theorem holds. Specifically, a Montel space is a barrelled topological vector space in which every closed and bounded subset is compact. Definition A topological vector space (TVS) has the if every closed and bounded subset is compact. A is a barrelled topological vector space with the Heine–Borel property. Equivalently, it is an infrabarrelled semi-Montel space where a Hausdorff locally convex topological vector space is called a or if every bounded subset is relatively compact.A subset S of a topological space X is called relatively compact is its closure in X is compact. A subset of a TVS is compact if and only if it is complete and totally bounded. A is a Fréchet space that is also a Montel space. Characterizations A separable Fréchet space is a Montel space if and only if each weak-* ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minkowski Space
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded. Although initially developed by mathematician Hermann Minkowski for Maxwell's equations of electromagnetism, the mathematical structure of Minkowski spacetime was shown to be implied by the postulates of special relativity. Minkowski space is closely associated with Einstein's theories of special relativity and general relativity and is the most common mathematical structure on which special relativity is formulated. While the individual components in Euclidean space and time may differ due to length contraction and time dilation, in Minkowski spacetime, all frames of reference will agree on the total distance in spacetime between events.This makes spacetime distance an invarian ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
LF-space
In mathematics, an ''LF''-space, also written (''LF'')-space, is a topological vector space (TVS) ''X'' that is a locally convex inductive limit of a countable inductive system (X_n, i_) of Fréchet spaces. This means that ''X'' is a direct limit of a direct system (X_n, i_) in the category of locally convex topological vector spaces and each X_n is a Fréchet space. The name ''LF'' stands for Limit of Fréchet spaces. If each of the bonding maps i_ is an embedding of TVSs then the ''LF''-space is called a strict ''LF''-space. This means that the subspace topology induced on by is identical to the original topology on . Some authors (e.g. Schaefer) define the term "''LF''-space" to mean "strict ''LF''-space," so when reading mathematical literature, it is recommended to always check how ''LF''-space is defined. Definition Inductive/final/direct limit topology Throughout, it is assumed that * \mathcal is either the category of topological spaces or some subcategory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
