The oriented derivative

TL;DR

This paper introduces an oriented derivative concept in Banach and Hilbert spaces, extending classical calculus rules and enabling additive decomposition in infinite-dimensional settings.

Contribution

It presents a novel perspective on derivatives as oriented towards star convex sets, generalizing fundamental calculus theorems in infinite-dimensional spaces.

Findings

Extends mean value theorem, chain rule, and Taylor formula to oriented derivatives.

Decomposes additively along orthogonal sums in Hilbert spaces.

Provides a new framework for differential calculus in Banach and Hilbert spaces.

Abstract

We show that the derivatives in the sense of Fr\'echet and G\^ateaux can be viewed as derivatives oriented towards a star convex set with the origin as center. The resulting oriented differential calculus extends the mean value theorem, the chain rule and the Taylor formula in Banach spaces. Moreover, the oriented derivative decomposes additively along countably infinite orthogonal sums in Hilbert spaces.