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In mathematics, a real or complex-valued function ''f'' on ''d''-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants ''C'', α, such that : for all ''x'' and ''y'' in the domain of ''f''. More generally, the condition can be formulated for functions between any two metric spaces. The number α is called the ''exponent'' of the Hölder condition. If α = 1, then the function satisfies a Lipschitz condition. If α = 0, then the function simply is bounded. The condition is named after Otto Hölder. We have the following chain of inclusions for functions over a compact subset of the real line : Continuously differentiable ⊆Lipschitz continuous ⊆ α-Hölder continuous ⊆ uniformly continuous ⊆ continuous where 0 < α ≤1. ==Hölder spaces== Hölder spaces consisting of functions satisfying a Hölder condition are basic in areas of functional analysis relevant to solving partial differential equations, and in dynamical systems. The Hölder space ''C''''k'',α(Ω), where Ω is an open subset of some Euclidean space and ''k'' ≥ 0 an integer, consists of those functions on Ω having continuous derivatives up to order ''k'' and such that the ''k''th partial derivatives are Hölder continuous with exponent α, where 0 < α ≤ 1. This is a locally convex topological vector space. If the Hölder coefficient : is finite, then the function ''f'' is said to be ''(uniformly) Hölder continuous with exponent α in Ω.'' In this case, the Hölder coefficient serves as a seminorm. If the Hölder coefficient is merely bounded on compact subsets of Ω, then the function ''f'' is said to be ''locally Hölder continuous with exponent α in Ω.'' If the function ''f'' and its derivatives up to order ''k'' are bounded on the closure of Ω, then the Hölder space can be assigned the norm : These norms and seminorms are often denoted simply and or also and in order to stress the dependence on the domain of ''f''. If Ω is open and bounded, then is a Banach space with respect to the norm . 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「Hölder condition」の詳細全文を読む スポンサード リンク
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