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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Hilbert–Schmidt theorem ： ウィキペディア英語版 Hilbert–Schmidt theorem In mathematical analysis, the Hilbert–Schmidt theorem, also known as the eigenfunction expansion theorem, is a fundamental result concerning compact, self-adjoint operators on Hilbert spaces. In the theory of partial differential equations, it is very useful in solving elliptic boundary value problems. ==Statement of the theorem== Let (''H'', ⟨ , ⟩) be a real or complex Hilbert space and let ''A'' : ''H'' → ''H'' be a bounded, compact, self-adjoint operator. Then there is a sequence of non-zero real eigenvalues ''λ''''i'', ''i'' = 1, ..., ''N'', with ''N'' equal to the rank of ''A'', such that |''λ''''i''| is monotonically non-increasing and, if ''N'' = +∞, :$\backslash lim\_\; \backslash lambda\_\; =\; 0.$ Furthermore, if each eigenvalue of ''A'' is repeated in the sequence according to its multiplicity, then there exists an orthonormal set ''φ''''i'', ''i'' = 1, ..., ''N'', of corresponding eigenfunctions, i.e. :$A\; \backslash varphi\_\; =\; \backslash lambda\_\; \backslash varphi\_\; \backslash mbox\; i\; =\; 1,\; \backslash dots,\; N.$ Moreover, the functions ''φ''''i'' form an orthonormal basis for the range of ''A'' and ''A'' can be written as :$A\; u\; =\; \backslash sum\_^\; \backslash lambda\_\; \backslash langle\; \backslash varphi\_,\; u\; \backslash rangle\; \backslash varphi\_\; \backslash mbox\; u\; \backslash in\; H.$ 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「Hilbert–Schmidt theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.