We discuss an alternative approach to Fréchet derivatives on Banach spaces inspired by a characterisation of derivatives due to Carathéodory. The approach allows many questions of differentiability to be reduced to questions of continuity. We demonstrate how that simplifies the theory of differentiation, including the rules of differentiation and the Schwarz lemma on the symmetry of second-order derivatives. We also provide a short proof of the differentiable dependence of fixed points in the Banach fixed point theorem.

中文翻译：

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FRÉCHET 衍生物的另一种方法

我们讨论了一种在 Banach 空间上的 Fréchet 导数的替代方法，其灵感来自于 Carathéodory 对导数的表征。该方法允许将许多可微性问题简化为连续性问题。我们展示了这如何简化微分理论，包括微分规则和关于二阶导数对称性的施瓦茨引理。我们还提供了 Banach 不动点定理中不动点可微依赖的简短证明。