About: Peter–Weyl theorem

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In mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily 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.

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rdfs:label	Satz von Peter-Weyl (de) Teorema de Peter-Weyl (es) Teorema di Peter-Weyl (it) 페터-바일 정리 (ko) Peter–Weyl theorem (en) Teorema de Peter-Weyl (pt) 彼得-魏尔定理 (zh)
rdfs:comment	Im mathematischen Teilgebiet der harmonischen Analyse verallgemeinert der Satz von Peter-Weyl, benannt nach Hermann Weyl und seinem Studenten Fritz Peter (1899–1949), die Fourierreihe für Funktionen auf beliebigen kompakten topologischen Gruppen. (de) 彼得-魏尔定理（英語：Peter–Weyl theorem）是调和分析和群表示论中的一组重要定理，于1927年由赫尔曼·魏尔和他的学生证明。该定理刻画了紧群不可约表示的完备性，可以视作有限群表示理论中弗罗贝尼乌斯定理的推广。定理分为三部分：第一部分指出，紧群的所有有限维不可约的，在上所有复值连续群函数构成、配备了的空间中稠密。第二部分指出，在任何一个可分希尔伯特空间上的酉表示都完全可约。第三部分断言，的所有有限维不可约酉表示的矩阵元构成了上平方可积的复值函数空间的一组标准正交基。 (zh) El Teorema de Peter-Weyl es un resultado básico en la teoría del análisis armónico, aplicado a grupos topológicos que son compactos, pero no necesariamente . Hermann Weyl, junto con su estudiante Peter, lo probó en la configuración de un grupo compacto de Lie, G. El teorema generaliza los hechos significantes sobre la descomposición de la representación regular de un grupo finito, como fue descubierto por F.G. Frobenius e Issai Schur. (es) In mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily 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. (en) Il teorema di Peter-Weyl è un risultato della teoria delle rappresentazioni che fornisce informazioni utili al calcolo delle rappresentazioni irriducibili di gruppi finiti (informazioni sul numero delle rappresentazioni irriducibili non equivalenti e sulla loro dimensione). Esso può anche essere usato per decomporre le rappresentazioni riducibili. L'uso di questo teorema per i gruppi finiti viene ulteriormente semplificato introducendo la nozione di carattere, e ne esiste inoltre una generalizzazione per rappresentazioni di gruppi infiniti come ad esempio i gruppi di Lie. (it)
dct:subject	Theorems in group theory Topological groups Unitary representation theory Theorems in harmonic analysis Theorems in representation theory
Wikipage page ID	294474 (xsd:integer)
Wikipage revision ID	1106662564 (xsd:integer)
Link from a Wikipage to another Wikipage	Annals of Mathematics Character (mathematics) Uniform norm Unitary operator Lie group Profinite group Continuous function Mathematics Closure (topology) Endomorphism Continuous functions on a compact Hausdorff space Homogeneous polynomials Limit (category theory) Lp space Stone–Weierstrass theorem Compact group Dense set Élie Cartan Fritz Peter Closed subgroup theorem Harmonic analysis Orthonormal basis Matrix coefficient Theorems in group theory Topological groups Topological group Unitary representation theory Haar measure Hausdorff space Irreducible representation Ferdinand Georg Frobenius Linear functional Regular representation Group action (mathematics) Hermann Weyl Issai Schur Matrix group Abelian group Maximal torus Theorems in harmonic analysis Theorems in representation theory Pontryagin duality Special unitary group Spherical harmonics Circle group Group representation Group homomorphisms Category theory Strong operator topology Unitary representation Complex-valued function Square-integrable function Finite group Weyl character formula Direct sum of representations dbr:American_Journal_of__Mathematics
sameAs	Peter–Weyl theorem Peter–Weyl theorem Peter–Weyl theorem Peter–Weyl theorem Peter–Weyl theorem Peter–Weyl theorem Peter–Weyl theorem Peter–Weyl theorem Peter–Weyl theorem
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