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Bilinear Map
In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example. Definition Vector spaces Let V, W and X be three vector spaces over the same base field F. A bilinear map is a function B : V \times W \to X such that for all w \in W, the map B_w v \mapsto B(v, w) is a linear map from V to X, and for all v \in V, the map B_v w \mapsto B(v, w) is a linear map from W to X. In other words, when we hold the first entry of the bilinear map fixed while letting the second entry vary, the result is a linear operator, and similarly for when we hold the second entry fixed. Such a map B satisfies the following properties. * For any \lambda \in F, B(\lambda v,w) = B(v, \lambda w) = \lambda B(v, w). * The map B is additive in both components: if v_1, v_2 \in V and w_1, w_2 \in W, then B(v_1 + v_2, w) = B(v_1, w) + B(v_2, w) and B(v, w_1 + w_2) = B(v ... [...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] |
Zero Vector
In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context. Additive identities An additive identity is the identity element in an additive group. It corresponds to the element 0 such that for all x in the group, . Some examples of additive identity include: * The zero vector under vector addition: the vector of length 0 and whose components are all 0. Often denoted as \mathbf or \vec. * The zero function or zero map defined by , under pointwise addition * The empty set under set union * An empty sum or empty coproduct * An initial object in a category (an empty coproduct, and so an identity under coproducts) Absorbing elements An absorbing element in a multiplicative semigroup or semiring generalises the property . Examples include: *The empty set, which is an absorbing element under Cartesian product of sets, since *The zero functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strong Dual
In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded subsets of X, where this topology is denoted by b\left(X^, X\right) or \beta\left(X^, X\right). The coarsest polar topology is called weak topology. The strong dual space plays such an important role in modern functional analysis, that the continuous dual space is usually assumed to have the strong dual topology unless indicated otherwise. To emphasize that the continuous dual space, X^, has the strong dual topology, X^_b or X^_ may be written. Strong dual topology Throughout, all vector spaces will be assumed to be over the field \mathbb of either the real numbers \R or complex numbers \C. Definition from a dual system Let (X, Y, \langle \cdot, \cdot \rangle) be a dual pair of vector spaces over the field \mathbb of real numbers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metrizable
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) such that the topology induced by d is \mathcal. Metrization theorems are theorems that give sufficient conditions for a topological space to be metrizable. Properties Metrizable spaces inherit all topological properties from metric spaces. For example, they are Hausdorff paracompact spaces (and hence normal and Tychonoff) and first-countable. However, some properties of the metric, such as completeness, cannot be said to be inherited. This is also true of other structures linked to the metric. A metrizable uniform space, for example, may have a different set of contraction maps than a metric space to which it is homeomorphic. Metrization theorems One of the first widely recognized metrization theorems was . This states that every H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Baire Space
In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior. According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are examples of Baire spaces. The Baire category theorem combined with the properties of Baire spaces has numerous applications in topology, geometry, analysis, in particular functional analysis. Bourbaki introduced the term "Baire space" in honor of René Baire, who investigated the Baire category theorem in the context of Euclidean space \R^n in his 1899 thesis. Definition The definition that follows is based on the notions of meagre (or first category) set (namely, a set that is a countable union of sets whose closure has empty interior) and nonmeagre (or second category) set (namely, a set that is not meagre). See the corresponding article for details. A topological space X is called a Baire space if it satisfies any of the follo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypocontinuity
In mathematics, a hypocontinuous is a condition on bilinear maps of topological vector spaces that is weaker than continuity but stronger than separate continuity. Many important bilinear maps that are not continuous are, in fact, hypocontinuous. Definition If X, Y and Z are topological vector spaces then a bilinear map \beta: X\times Y\to Z is called ''hypocontinuous'' if the following two conditions hold: * for every bounded set A\subseteq X the set of linear maps \ is an equicontinuous subset of Hom(Y,Z), and * for every bounded set B\subseteq Y the set of linear maps \ is an equicontinuous subset of Hom(X,Z). Sufficient conditions Theorem: Let ''X'' and ''Y'' be barreled spaces and let ''Z'' be a locally convex space. Then every separately continuous bilinear map of X \times Y into ''Z'' is hypocontinuous. Examples * If ''X'' is a Hausdorff locally convex barreled space In functional analysis and related areas of mathematics, a barrelled space (also written barreled spac ... [...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] |
Cross Product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is denoted by the symbol \times. Given two linearly independent vectors and , the cross product, (read "a cross b"), is a vector that is perpendicular to both and , and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with the dot product (projection product). If two vectors have the same direction or have the exact opposite direction from each other (that is, they are ''not'' linearly independent), or if either one has zero length, then their cross product is zero. More generally, the magnitude of the product equals the area of a parallelogram with the vectors for sides; in particular, the magnitude of the product of two perpendic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by constants. The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the . When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called the ''continuous dual space''. Dual vector spaces find application in many branches of mathematics that use vector spaces, such as in tensor analysis with finite-dimensional vector spaces. When applied to vector spaces of functions (which are typically infinite-dimensional), dual spaces are used to describe measures, distributions, and Hilbert spaces. Consequently, the dual space is an important concept in functional analysis. Early terms for ''dual'' include ''polarer Raum'' ahn 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inner Product Space
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often denoted with angle brackets such as in \langle a, b \rangle. Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors. Inner product spaces generalize Euclidean vector spaces, in which the inner product is the dot product or ''scalar product'' of Cartesian coordinates. Inner product spaces of infinite dimension are widely used in functional analysis. Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces. The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in 1898. An inner product naturally induces an associated norm, (denoted , x, and , y, in the picture); so, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers is denoted or \mathbb and is sometimes called "the reals". The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality. The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix (mathematics)
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, \begin1 & 9 & -13 \\20 & 5 & -6 \end is a matrix with two rows and three columns. This is often referred to as a "two by three matrix", a "-matrix", or a matrix of dimension . Without further specifications, matrices represent linear maps, and allow explicit computations in linear algebra. Therefore, the study of matrices is a large part of linear algebra, and most properties and operations of abstract linear algebra can be expressed in terms of matrices. For example, matrix multiplication represents composition of linear maps. Not all matrices are related to linear algebra. This is, in particular, the case in graph theory, of incidence matrices, and adjacency matrices. ''This article focuses on matrices related to linear algebra, and, unle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
