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Polynomially Reflexive Space
In mathematics, a polynomially reflexive space is a Banach space ''X'', on which the space of all polynomials in each degree is a reflexive space. Given a multilinear functional ''M''''n'' of degree ''n'' (that is, ''M''''n'' is ''n''-linear), we can define a polynomial ''p'' as :p(x)=M_n(x,\dots,x) (that is, applying ''M''''n'' on the ''diagonal'') or any finite sum of these. If only ''n''-linear functionals are in the sum, the polynomial is said to be ''n''-homogeneous. We define the space ''P''''n'' as consisting of all ''n''-homogeneous polynomials. The ''P''1 is identical to the dual space, and is thus reflexive for all reflexive ''X''. This implies that reflexivity is a prerequisite for polynomial reflexivity. Relation to continuity of forms On a finite-dimensional linear space, a quadratic form ''x''↦''f''(''x'') is always a (finite) linear combination of products ''x''↦''g''(''x'') ''h''(''x'') of two linear functionals ''g'' and ''h''. Therefore, assuming that th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Orthonormal
In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal unit vectors. A unit vector means that the vector has a length of 1, which is also known as normalized. Orthogonal means that the vectors are all perpendicular to each other. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of unit length. An orthonormal set which forms a basis is called an ''orthonormal basis''. Intuitive overview The construction of orthogonality of vectors is motivated by a desire to extend the intuitive notion of perpendicular vectors to higher-dimensional spaces. In the Cartesian plane, two vectors are said to be ''perpendicular'' if the angle between them is 90° (i.e. if they form a right angle). This definition can be formalized in Cartesian space by defining the dot product and specifying that two vectors in the plane are orthogonal if their dot product is zero. Similarly, the construction of the norm of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 |
Quotient Space (linear Algebra)
In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by "collapsing" N to zero. The space obtained is called a quotient space and is denoted V/N (read "V mod N" or "V by N"). Definition Formally, the construction is as follows. Let V be a vector space over a field \mathbb, and let N be a subspace of V. We define an equivalence relation \sim on V by stating that x \sim y iff . That is, x is related to y if and only if one can be obtained from the other by adding an element of N. This definition implies that any element of N is related to the zero vector; more precisely, all the vectors in N get mapped into the equivalence class of the zero vector. The equivalence class – or, in this case, the coset – of x is defined as : := \ and is often denoted using the shorthand = x + N. The quotient space V/N is then defined as V/_\sim, the set of all equivalence classes induced by \sim on V. Scalar multiplication and addition are defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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. Preliminaries The -norm in finite dimensions The Euclidean length of a vector x = (x_1, x_2, \dots, x_n) in the n-dimensional real vector space \Reals^n is given by the Euclidean norm: \, x\, _2 = \left(^2 + ^2 + \dotsb + ^2\right)^. The Euclidean distance between two points x and y is the length \, x - y\, _2 of the straight line b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Approximation Property
In mathematics, specifically functional analysis, a Banach space is said to have the approximation property (AP), if every compact operator is a limit of finite-rank operators. The converse is always true. Every Hilbert space has this property. There are, however, Banach spaces which do not; Per Enflo published the first counterexample in a 1973 article. However, much work in this area was done by Grothendieck (1955). Later many other counterexamples were found. The space \mathcal L(H) of bounded operators on an infinite-dimensional Hilbert space H does not have the approximation property. The spaces \ell^p for p\neq 2 and c_0 (see Sequence space) have closed subspaces that do not have the approximation property. Definition A locally convex topological vector space ''X'' is said to have the approximation property, if the identity map can be approximated, uniformly on precompact sets, by continuous linear maps of finite rank. For a locally convex space ''X'', the followin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Reflexive Space
In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space for which the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is a homeomorphism (or equivalently, a TVS isomorphism). A normed space is reflexive if and only if this canonical evaluation map is surjective, in which case this (always linear) evaluation map is an isometric isomorphism and the normed space is a Banach space. Those spaces for which the canonical evaluation map is surjective are called semi-reflexive spaces. In 1951, R. C. James discovered a Banach space, now known as James' space, that is reflexive (meaning that the canonical evaluation map is not an isomorphism) but is nevertheless isometrically isomorphic to its bidual (any such isometric isomorphism is necessarily the canonical evaluation map). So importantly, for a Banach space to be reflexive, it is not enough for it to be isometri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Continuous Function (topology)
In mathematics, a continuous function is a function (mathematics), function such that a small variation of the argument of a function, argument induces a small variation of the Value (mathematics), value of the function. This implies there are no abrupt changes in value, known as ''Classification of discontinuities, discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Until the 19th century, mathematicians largely relied on Intuition, intuitive notions of continuity and considered only continuous functions. The (ε, δ)-definition of limit, epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real number, real and complex number, complex numbers. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Weak Convergence (Hilbert Space)
In mathematics, weak convergence in a Hilbert space is the convergence of a sequence of points in the weak topology. Definition A sequence of points (x_n) in a Hilbert space ''H'' is said to converge weakly to a point ''x'' in ''H'' if :\lim_\langle x_n,y \rangle = \langle x,y \rangle for all ''y'' in ''H''. Here, \langle \cdot, \cdot \rangle is understood to be the inner product on the Hilbert space. The notation :x_n \rightharpoonup x is sometimes used to denote this kind of convergence. Properties *If a sequence converges strongly (that is, if it converges in norm), then it converges weakly as well. *Since every closed and bounded set is weakly relatively compact (its closure in the weak topology is compact), every bounded sequence x_n in a Hilbert space ''H'' contains a weakly convergent subsequence. Note that closed and bounded sets are not in general weakly compact in Hilbert spaces (consider the set consisting of an orthonormal basis in an infinite-dimensional Hilbert ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Hilbert Space
In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The inner product allows lengths and angles to be defined. Furthermore, Complete metric space, completeness means that there are enough limit (mathematics), limits in the space to allow the techniques of calculus to be used. A Hilbert space is a special case of a Banach space. Hilbert spaces were studied beginning in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, mathematical formulation of quantum mechanics, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Banach Space
In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term " Fréchet space". Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces. Definition A Banach space is a complete nor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Linear Functional
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear mapIn some texts the roles are reversed and vectors are defined as linear maps from covectors to scalars from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the set of all linear functionals from to is itself a vector space over with addition and scalar multiplication defined pointwise. This space is called the dual space of , or sometimes the algebraic dual space, when a topological dual space is also considered. It is often denoted , p. 19, §3.1 or, when the field is understood, V^*; other notations are also used, such as V', V^ or V^. When vectors are represented by column vectors (as is common when a basis is fixed), then linear functionals are represented as row vectors, and their values on specific vectors are given by matrix products (with the row vector on the left). Examples T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
