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In mathematics, the Gâteaux differential or Gâteaux derivative is a generalization of the concept of directional derivative in differential calculus. Named after René Gâteaux, a French mathematician who died young in World War I, it is defined for functions between locally convex topological vector spaces such as Banach spaces. Like the Fréchet derivative on a Banach space, the Gâteaux differential is often used to formalize the functional derivative commonly used in the calculus of variations and physics. Unlike other forms of derivatives, the Gâteaux differential of a function may be nonlinear. However, often the definition of the Gâteaux differential also requires that it be a continuous linear transformation. Some authors, such as , draw a further distinction between the Gâteaux differential (which may be nonlinear) and the Gâteaux derivative (which they take to be linear). In most applications, continuous linearity follows from some more primitive condition which is natural to the particular setting, such as imposing complex differentiability in the context of infinite dimensional holomorphy or continuous differentiability in nonlinear analysis. ==Definition== Suppose ''X'' and ''Y'' are locally convex topological vector spaces (for example, Banach spaces), ''U'' ⊂ ''X'' is open, and ''F'' : ''X'' → ''Y''. The Gâteaux differential ''dF''(''u'';''ψ'') of ''F'' at ''u'' ∈ ''U'' in the direction ''ψ'' ∈ ''X'' is defined as if the limit exists. If the limit exists for all ''ψ'' ∈ ''X'', then one says that ''F'' is Gâteaux differentiable at ''u''. The limit appearing in () is taken relative to the topology of ''Y''. If ''X'' and ''Y'' are real topological vector spaces, then the limit is taken for real τ. On the other hand, if ''X'' and ''Y'' are complex topological vector spaces, then the limit above is usually taken as τ→0 in the complex plane as in the definition of complex differentiability. In some cases, a weak limit is taken instead of a strong limit, which leads to the notion of a weak Gâteaux derivative. 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「Gâteaux derivative」の詳細全文を読む スポンサード リンク
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