I have posted the following question (with minimal differences) on MSE some days ago, without receiving a satisfactory answer, so let me try here again.

Take the vector space of infinitely differentiable functions on $[0,1]$. The standard norm of $C^k([0,1])$ is just the $\ell^1$-norm of the vector $(\|f\|_\infty, \|f'\|_\infty,\ldots,\|f^{(k)}\|_\infty)$, but of course this idea cannot be further pursued to define a norm on $C^\infty([0,1])$.

However, what if one would consider the space $$ \mathcal S_p:=\{f\in C^\infty([0,1]):(\|f^{(n)}\|_\infty)_{n\in\mathbb N}\in \ell^p \} $$ for $p\in [1,\infty]$? These spaces are certainly small - in particular, for $p<\infty$ $\mathcal S_p$ contains neither $\exp$, nor $\sin$ and $\cos$ - but at least they do contain the polynomials and they seem to be Banach spaces - in fact even Banach lattice algebras. Do these spaces appear in applications, especially in PDEs? Has anybody ever studied their functional analytical properties and if this is not the case, what are these spaces' most obvious drawbacks?

(@JochenWengenroth has already pointed out in a comment on MSE that partition of unity would not hold in these spaces.)