Is the Borel lemma projection a smooth principal bundle?

Question

Consider the Fréchet spaces $C^\infty(\mathbb{R},\mathbb{R})$ and $\mathbb{R}^\infty$, and the continuous linear map $$ J\colon C^\infty(\mathbb{R},\mathbb{R}) \to \mathbb{R}^\infty $$ returning the infinite jet at 0, which is a surjection by Borel's lemma. Here $\mathbb{R}^\infty$ is the set of all sequences of real numbers, with the family of seminorms induced by the truncations to the first $n$-coordinates. The map $J$ does not have a continuous linear section, but it has a continuous non-linear section. What I'd like to know is if $J$ has a smooth nonlinear section, even if just in a neighbourhood of $0$. Here smoothness is taken in the sense of Michal–Bastiani.

I recently learned that smooth maps $\mathbb{R}^n \to \mathbb{R}^\infty$ lift smoothly to $C^\infty(\mathbb{R},\mathbb{R})$ (via Enxin Wu's paper Homological Algebra for Diffeological Vector Spaces), which makes this projection maps a subduction of the associated diffeological spaces, making our friend $J$ above a diffeological principal bundle (which, I remind you, are not assumed locally trivial!). But I do wonder if it's a bundle in the traditional sense, in the category of Fréchet manifolds. Perhaps a continuous section could be smoothed, but I really am grasping at straws.

Answer 1

No, there is not even any $C^1$ (in the Michal−Bastiani, i.e. Keller $C_c$ sence) map $\mathbb R^\infty=G\sqsupseteq{\rm dom}\,f\to E=C^\infty(\mathbb R)=C^\infty(\mathbb R,\mathbb R)$ with ${\rm dom}\,f$ a zero neighbourhood and $J\circ f={\rm id}$ on ${\rm dom}\,f$. The argument goes as follows. Supposing there is, for $j={\rm D}\,f(0)$ we have $J\circ j={\rm id}$ on $G$. Then for $F$ the subspace of $E$ formed by functions that are infinitely flat at $0$, and for $\rho:E\to F$ given by $x\mapsto x-j\circ J\,x$ we have $\rho$ the identity on $F$ which is shown to be impossible in Corollary 7.1.3 on page 206 in the Frölicher−Kriegl book Linear Spaces and Differentiation Theory. Note that since all spaces here are Fréchet, there is only one reasonable concept of smoothness that (in this restricted case) also equals that in the FK book. A $\underline{\rm Con}$−morphism (there and here; in this case) just means a continuous linear map.