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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 Riesz representation theorem ： ウィキペディア英語版 Riesz representation theorem There are several well-known theorems in functional analysis known as the Riesz representation theorem. They are named in honour of Frigyes Riesz. This article will describe his theorem concerning the dual of a Hilbert space, which is sometimes called the Fréchet-Riesz theorem. For the theorems relating linear functionals to measures, see Riesz–Markov–Kakutani representation theorem. == The Hilbert space representation theorem == This theorem establishes an important connection between a Hilbert space and its (continuous) dual space. If the underlying field is the real numbers, the two are isometrically isomorphic; if the underlying field is the complex numbers, the two are isometrically anti-isomorphic. The (anti-) isomorphism is a particular, natural one as will be described next. Let ''H'' be a Hilbert space, and let ''H *'' denote its dual space, consisting of all continuous linear functionals from ''H'' into the field R or C. If ''x'' is an element of ''H'', then the function φ''x'', for all ''y'' in ''H'' defined by :$\backslash varphi\_x(y)\; =\; \backslash left\backslash langle\; y\; ,\; x\; \backslash right\backslash rangle,$ where $\backslash langle\backslash cdot,\backslash cdot\backslash rangle$ denotes the inner product of the Hilbert space, is an element of ''H *''. The Riesz representation theorem states that ''every'' element of ''H *'' can be written uniquely in this form. Theorem. The mapping $\backslash Phi$: ''H'' → ''H *'' defined by $\backslash Phi(x)$ = $\backslash varphi\_x$ is an isometric (anti-) isomorphism, meaning that: * $\backslash Phi$ is bijective. * The norms of $x$ and $\backslash varphi\_x$ agree: $\backslash Vert\; x\; \backslash Vert\; =\; \backslash Vert\backslash Phi(x)\backslash Vert$. * $\backslash Phi$ is additive: $\backslash Phi(\; x\_1\; +\; x\_2\; )\; =\; \backslash Phi(\; x\_1\; )\; +\; \backslash Phi(\; x\_2\; )$. * If the base field is R, then $\backslash Phi(\backslash lambda\; x)\; =\; \backslash lambda\; \backslash Phi(x)$ for all real numbers λ. * If the base field is C, then $\backslash Phi(\backslash lambda\; x)\; =\; \backslash bar\; \backslash Phi(x)$ for all complex numbers λ, where $\backslash bar$ denotes the complex conjugation of λ. The inverse map of $\backslash Phi$ can be described as follows. Given a non-zero element $\backslash varphi$ of ''H *'', the orthogonal complement of the kernel of $\backslash varphi$ is a one-dimensional subspace of ''H''. Take a non-zero element ''z'' in that subspace, and set $x\; =\; \backslash overline\; \backslash cdot\; z\; /^2$. Then $\backslash Phi(x)$ = $\backslash varphi$. Historically, the theorem is often attributed simultaneously to Riesz and Fréchet in 1907 (see references). In the mathematical treatment of quantum mechanics, the theorem can be seen as a justification for the popular bra–ket notation. When the theorem holds, every ket $|\backslash psi\backslash rangle$ has a corresponding bra $\backslash langle\backslash psi|$, and the correspondence is unambiguous. cf. also Rigged Hilbert space 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「Riesz representation 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.