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

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