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In mathematics, the quasi-derivative is one of several generalizations of the derivative of a function between two Banach spaces. The quasi-derivative is a slightly stronger version of the Gâteaux derivative, though weaker than the Fréchet derivative. Let ''f'' : ''A'' → ''F'' be a continuous function from an open set ''A'' in a Banach space ''E'' to another Banach space ''F''. Then the quasi-derivative of ''f'' at ''x''0 ∈ ''A'' is a linear transformation ''u'' : ''E'' → ''F'' with the following property: for every continuous function ''g'' : () → ''A'' with ''g''(0)=''x''0 such that ''g''′(0) ∈ ''E'' exists, : If such a linear map ''u'' exists, then ''f'' is said to be ''quasi-differentiable'' at ''x''0. Continuity of ''u'' need not be assumed, but it follows instead from the definition of the quasi-derivative. If ''f'' is Fréchet differentiable at ''x''0, then by the chain rule, ''f'' is also quasi-differentiable and its quasi-derivative is equal to its Fréchet derivative at ''x''0. The converse is true provided ''E'' is finite-dimensional. Finally, if ''f'' is quasi-differentiable, then it is Gâteaux differentiable and its Gâteaux derivative is equal to its quasi-derivative. ==References== * 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「Quasi-derivative」の詳細全文を読む スポンサード リンク
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