An alternative approach to Fr\'echet derivatives

TL;DR

This paper introduces a novel approach to Fréchet derivatives on Banach spaces, simplifying differentiation theory by linking it to continuity and providing new proofs for classical results.

Contribution

It presents an alternative characterization of Fréchet derivatives inspired by Carathéodory, reducing differentiability questions to continuity and simplifying key differentiation rules.

Findings

Simplifies the theory of differentiation on Banach spaces

Provides a new proof of differentiable dependence of fixed points

Clarifies symmetry of second order derivatives

Abstract

We discuss an alternative approach to Fr\'echet derivatives on Banach spaces inspired by a characterisation of derivatives due to Carath\'eodory. The approach allows us to reduce many questions of differentiability to a question 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.