The Frolicher--Kriegl differentiabilities as a particular case of the Bertram--Glockner--Neeb construction

TL;DR

This paper demonstrates that the differentiability classes defined by Frolicher and Kriegl can be derived as special cases of the more general Bertram-Glockner-Neeb construction, unifying different approaches to differentiability.

Contribution

It shows that Frolicher-Kriegl differentiability classes are specific instances within the broader Bertram-Glockner-Neeb framework, connecting two theories of differentiability.

Findings

Frolicher-Kriegl differentiability classes are special cases of Bertram-Glockner-Neeb construction.

The paper unifies different notions of differentiability under a common framework.

Provides a general method to derive $C^k$ differentiability from $C^0$ concepts.

Abstract

We prove that the order $k$ differentiability classes for $k=0,1,...\infty$ in the "arc-generated" interpretation of the Lipschitz theory of differentiation by Frolicher and Kriegl can be obtained as particular cases of the general construction by Bertram, Glockner and Neeb leading to $C^k$ differentiabilities from a given $C^0$ concept.