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Peter–Weyl Theorem
In mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are Compact group, compact, but are not necessarily Abelian group, abelian. It was initially proved by Hermann Weyl, with his student Fritz Peter, in the setting of a compact topological group ''G'' . The theorem is a collection of results generalizing the significant facts about the decomposition of the regular representation of any finite group, as discovered by Ferdinand Georg Frobenius and Issai Schur. Let ''G'' be a compact group. The theorem has three parts. The first part states that the matrix coefficients of irreducible representations of ''G'' are dense in the space ''C''(''G'') of continuous complex-valued functions on ''G'', and thus also in the space ''L''2(''G'') of square-integrable functions. The second part asserts the complete reducibility of unitary representations of ''G''. The third part then asserts that the regular representati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Matrix Coefficient
In mathematics, a matrix coefficient (or matrix element) is a function on a group of a special form, which depends on a linear representation of the group and additional data. Precisely, it is a function on a compact topological group ''G'' obtained by composing a representation of ''G'' on a vector space ''V'' with a linear map from the endomorphisms of ''V'' into ''V'' underlying field. It is also called a representative function. They arise naturally from finite-dimensional representations of ''G'' as the matrix-entry functions of the corresponding matrix representations. The Peter–Weyl theorem says that the matrix coefficients on ''G'' are dense in the Hilbert space of square-integrable functions on ''G''. Matrix coefficients of representations of Lie groups turned out to be intimately related with the theory of special functions, providing a unifying approach to large parts of this theory. Growth properties of matrix coefficients play a key role in the classification ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circle Group
In mathematics, the circle group, denoted by \mathbb T or , is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers \mathbb T = \. The circle group forms a subgroup of , the multiplicative group of all nonzero complex numbers. Since \C^\times is abelian, it follows that \mathbb T is as well. A unit complex number in the circle group represents a rotation of the complex plane about the origin and can be parametrized by the angle measure : \theta \mapsto z = e^ = \cos\theta + i\sin\theta. This is the exponential map for the circle group. The circle group plays a central role in Pontryagin duality and in the theory of Lie groups. The notation \mathbb T for the circle group stems from the fact that, with the standard topology (see below), the circle group is a 1-torus. More generally, \mathbb T^n (the direct product of \mathbb T with itself n times) is geometrically an n-toru ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Action (mathematics)
In mathematics, a group action of a group G on a set (mathematics), set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformation (function), transformations form a group (mathematics), group under function composition; for example, the rotation (mathematics), rotations around a point in the plane. It is often useful to consider the group as an abstract group, and to say that one has a group action of the abstract group that consists of performing the transformations of the group of transformations. The reason for distinguishing the group from the transformations is that, generally, a group of transformations of a mathematical structure, structure acts also on various related structures; for example, the above rotation group also acts on triangles by transforming triangles into triangles. If a group acts on a structure, it will usually also act on objects built from that st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (to allow division), or equivalently, the concept of addition and subtraction. Combining these two ideas, one obtains a continuous group where multiplying points and their inverses is continuous. If the multiplication and taking of inverses are smoothness, smooth (differentiable) as well, one obtains a Lie group. Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the circle group. Rotating a circle is an example of a continuous symmetry. For an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Group
In mathematics, a matrix group is a group ''G'' consisting of invertible matrices over a specified field ''K'', with the operation of matrix multiplication. A linear group is a group that is isomorphic to a matrix group (that is, admitting a faithful, finite-dimensional representation over ''K''). Any finite group is linear, because it can be realized by permutation matrices using Cayley's theorem. Among infinite groups, linear groups form an interesting and tractable class. Examples of groups that are not linear include groups which are "too big" (for example, the group of permutations of an infinite set), or which exhibit some pathological behavior (for example, finitely generated infinite torsion groups). Definition and basic examples A group ''G'' is said to be ''linear'' if there exists a field ''K'', an integer ''d'' and an injective homomorphism from ''G'' to the general linear group GL''d''(''K'') (a faithful linear representation of dimension ''d'' over ''K''): i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stone–Weierstrass Theorem
In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval (mathematics), interval can be uniform convergence, uniformly approximated as closely as desired by a polynomial function. Because polynomials are among the simplest functions, and because computers can directly evaluate polynomials, this theorem has both practical and theoretical relevance, especially in polynomial interpolation. The original version of this result was established by Karl Weierstrass in #Historical works, 1885 using the Weierstrass transform. Marshall H. Stone considerably generalized the theorem and simplified the proof. His result is known as the Stone–Weierstrass theorem. The Stone–Weierstrass theorem generalizes the Weierstrass approximation theorem in two directions: instead of the real interval , an arbitrary compact space, compact Hausdorff space is considered, and instead of the Algebra over a field, algebra of polynomial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Norm
In mathematical analysis, the uniform norm (or ) assigns, to real- or complex-valued bounded functions defined on a set , the non-negative number :\, f\, _\infty = \, f\, _ = \sup\left\. This norm is also called the , the , the , or, when the supremum is in fact the maximum, the . The name "uniform norm" derives from the fact that a sequence of functions converges to under the metric derived from the uniform norm if and only if converges to uniformly. If is a continuous function on a closed and bounded interval, or more generally a compact set, then it is bounded and the supremum in the above definition is attained by the Weierstrass extreme value theorem, so we can replace the supremum by the maximum. In this case, the norm is also called the . In particular, if is some vector such that x = \left(x_1, x_2, \ldots, x_n\right) in finite dimensional coordinate space, it takes the form: :\, x\, _\infty := \max \left(\left, x_1\ , \ldots , \left, x_n\\right). This is c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Functions On A Compact Hausdorff Space
In mathematical analysis, and especially functional analysis, a fundamental role is played by the space of continuous functions on a compact Hausdorff space X with values in the real or complex numbers. This space, denoted by \mathcal(X), is a vector space with respect to the pointwise addition of functions and scalar multiplication by constants. It is, moreover, a normed space with norm defined by \, f\, = \sup_ , f(x), , the uniform norm. The uniform norm defines the topology of uniform convergence of functions on X. The space \mathcal(X) is a Banach algebra with respect to this norm. Properties * By Urysohn's lemma, \mathcal(X) separates points of X: If x, y \in X are distinct points, then there is an f \in \mathcal(X) such that f(x) \neq f(y). * The space \mathcal(X) is infinite-dimensional whenever X is an infinite space (since it separates points). Hence, in particular, it is generally not locally compact. * The Riesz–Markov–Kakutani representation theorem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dense Set
In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation). Formally, A is dense in X if the smallest closed subset of X containing A is X itself. The of a topological space X is the least cardinality of a dense subset of X. Definition A subset A of a topological space X is said to be a of X if any of the following equivalent conditions are satisfied: The smallest closed subset of X containing A is X itself. The closure of A in X is equal to X. That is, \operatorname_X A = X. The interior of the complement of A is empty. That is, \operatorname_X (X \setminus A) = \varnothing. Every point in X eith ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a group is a group homomorphism . In general, we can talk about endomorphisms in any category. In the category of sets, endomorphisms are functions from a set ''S'' to itself. In any category, the composition of any two endomorphisms of is again an endomorphism of . It follows that the set of all endomorphisms of forms a monoid, the full transformation monoid, and denoted (or to emphasize the category ). Automorphisms An invertible endomorphism of is called an automorphism. The set of all automorphisms is a subset of with a group structure, called the automorphism group of and denoted . In the following diagram, the arrows denote implication: Endomorphism rings Any two endomorphisms of an abelian group, , can be ad ... [...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 (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called scalar (mathematics), ''scalars''. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. Real vector spaces and complex vector spaces are kinds of vector spaces based on different kinds of scalars: real numbers and complex numbers. Scalars can also be, more generally, elements of any field (mathematics), field. Vector spaces generalize Euclidean vectors, which allow modeling of Physical quantity, physical quantities (such as forces and velocity) that have not only a Magnitude (mathematics), magnitude, but also a Orientation (geometry), direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix (mathematics), matrices, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
