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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 Banach–Stone theorem ： ウィキペディア英語版 Banach–Stone theorem In mathematics, the Banach–Stone theorem is a classical result in the theory of continuous functions on topological spaces, named after the mathematicians Stefan Banach and Marshall Stone. In brief, the Banach–Stone theorem allows one to recover a compact Hausdorff space from the algebra of scalars (the bounded continuous functions on the space). In modern language, this is the commutative case of the spectrum of a C *-algebra, and the Banach–Stone theorem can be seen as a functional analysis analog of the connection between a ring ''R'' and the spectrum of a ring Spec(''R'') in algebraic geometry. ==Statement of the theorem== For a topological space ''X'', let ''C''b(''X''; R) denote the normed vector space of continuous, real-valued, bounded functions ''f'' : ''X'' → R equipped with the supremum norm ||·||∞. This is an algebra, called the ''algebra of scalars'', under pointwise multiplication of functions. For a compact space ''X'', ''C''b(''X''; R) is the same as ''C''(''X''; R), the space of all continuous functions ''f'' : ''X'' → R. The algebra of scalars is a functional analysis analog of the ring of regular functions in algebraic geometry, there denoted $\backslash mathcal\_X$. Let ''X'' and ''Y'' be compact, Hausdorff spaces and let ''T'' : ''C''(''X''; R) → ''C''(''Y''; R) be a surjective linear isometry. Then there exists a homeomorphism ''φ'' : ''Y'' → ''X'' and ''g'' ∈ ''C''(''Y''; R) with :$|\; g(y)\; |\; =\; 1\; \backslash mbox\; y\; \backslash in\; Y$ and :$(T\; f)\; (y)\; =\; g(y)\; f(\backslash varphi(y))\; \backslash mbox\; y\; \backslash in\; Y,\; f\; \backslash in\; C(X;\; \backslash mathbf).$ 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「Banach–Stone 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.