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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 . 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 : and : 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
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