A universal approximation theorem and its applications to vector lattice theory

A classical result in approximation theory states that for any continuous function $\phi :\mathbb{R}\to \mathbb{R}$, the set $\mathrm{span}\{\phi \circ g:g\in \mathrm{Aff}(\mathbb{R})\}$ is dense in $\mathcal{C}(\mathbb{R})$ if and only if $\phi$ is not a polynomial. In this note, we present infinite dimensional variants of this result. These extensions apply to neural network architectures and improve the main density result obtained in \cite{BDG23}. We also discuss applications and related approximation results in vector lattices, improving and complementing results from \cite{AT:17, bhp,BT:24}.