[[!redirects Fréchet+manifolds+as+diffeological+spaces]] [[!redirects Fr\'echet+manifolds+as+diffeological+spaces]] * {: .title } Fréchet manifolds as diffeological spaces * {: .author } Losik, M. V. * {: .journal } Izv. Vyssh. Uchebn. Zaved. Mat. * {: .mrclass } 58B20 * {: .mrnumber } MR1213569 (94c:58008) * {: .pages } 36-42 * {: .year } 1992 {: .bibliography } --- ###Remarks### At the start of section 4, the article says (emphasis added): > By Fréchet manifolds we mean a $C^\infty$ manifold modeled on Fréchet space, which is defined in the standard manner by means of any definition of differentiability, **because all of them are equivalent**, as it has been demonstrated in [section] 3. But in the _Historical remarks_ at the end of chapter 1 of [[a convenient setting for global analysis]], it is said that there are 3 inequivalent notions of "infinitely differentiable" for Fréchet spaces. Digging a little deeper, my guess is that the different types of differentiability referred to in this article are all variants of the idea that a function $f \colon E \to F$ is differentiable if it is _directionally differentiable_, i.e., all directional derivatives exist. The variants are presumably related to how continuous the resulting directional derivatives are in both variables. Checking this will involve tracking down the references. --- category: smootheology [[!redirects Frechet manifolds as diffeological spaces]] [[!redirects frechet manifolds as diffeological spaces]] [[!redirects Fréchet manifolds as diffeological spaces]] [[!redirects fréchet manifolds as diffeological spaces]]