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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 F. G. Frobenius and Issai Schur. 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 representation of ''G'' on ''L''2(''G'') decomposes as the direct sum of all irreducible unitary representations. Moreover, the matrix coefficients of the irreducible unitary representations form an orthonormal basis of ''L''2(''G''). ==Matrix coefficients== A matrix coefficient of the group ''G'' is a complex-valued function φ on ''G'' given as the composition : where π : ''G'' → GL(''V'') is a finite-dimensional (continuous) group representation of ''G'', and ''L'' is a linear functional on the vector space of endomorphisms of ''V'' (e.g. trace), which contains GL(''V'') as an open subset. Matrix coefficients are continuous, since representations are by definition continuous, and linear functionals on finite-dimensional spaces are also continuous. The first part of the Peter–Weyl theorem asserts (; ): Peter-Weyl Theorem (Part I). The set of matrix coefficients of ''G'' is dense in the space of continuous complex functions C(''G'') on ''G'', equipped with the uniform norm. This first result resembles the Stone-Weierstrass theorem in that it indicates the density of a set of functions in the space of all continuous functions, subject only to an ''algebraic'' characterization. In fact, the matrix coefficients of tensor product form a unital algebra invariant under complex conjugation because the product of two matrix coefficients is a matrix coefficient of the tensor product representation, and the complex conjugate is a matrix coefficient of the dual representation. Hence the theorem follows directly from the Stone-Weierstrass theorem if the matrix coefficients separate points, which is obvious if ''G'' is a matrix group . Conversely, it is a consequence of the theorem that any compact Lie group is isomorphic to a matrix group . A corollary of this result is that the matrix coefficients of ''G'' are dense in ''L''2(''G''). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Peter–Weyl theorem」の詳細全文を読む スポンサード リンク
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