Seip's differentiability concepts as a particular case of the Bertram--Gloeckner--Neeb construction

TL;DR

This paper unifies Seip's differentiability with the Bertram--Gloeckner--Neeb framework, providing detailed proofs, examples, and set-theoretic reformulations to enhance understanding of differentiation theories.

Contribution

It demonstrates that Seip's differentiability concepts are a special case within the BGN construction, and discusses the practical limitations of Seip's inverse and implicit function theorems.

Findings

Seip's differentiability fits into the BGN framework.

Examples show limitations of Seip's theorems.

Reformulation aligns with set-theoretic approaches.

Abstract

From the point of view of unification of differentiation theory, it is of interest to note that the general construction principle of Bertram, Gloeckner and Neeb leading to a C^k differentiability concept from a given C^0 one, besides subsuming the Keller--Bastiani C_c^k differentiabilities on real Hausdorff locally convex spaces, also does the same to the "arc-generated" interpretation of the Lipschitz theory of differentiation by Frolicher and Kriegl, and likewise to the "compactly generated" theory of Seip's continuous differentiabilities. In this article, we give the details of the proof for the assertion concerning Seip's theory. We also give an example indicating that the premises in Seip's various inverse and implicit function theorems may be too strong in order for these theorems to have much practical value. Also included is a presentation of the BGN--setting reformulated so as to be consistent with the Kelley--Morse--Godel--Bernays--von Neumann type approach to set theory, as well as a treatment of the function space constructions and development of their basic properties needed in the proof of the main result.