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](https://ncatlab.org/nlab/show/Borel%27s+theorem). 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](https://ncatlab.org/nlab/show/Michal-Bastiani+smooth+map).


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](https://arxiv.org/abs/1406.6717)*), which makes this projection maps a *subduction* of the [associated diffeological spaces](https://ncatlab.org/nlab/show/diffeological+space#RelationBetweenDeffeologicalAndFrechetStructure), 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.



Is the Borel lemma projection a smooth principal bundle?
**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.


