A universal approximation theorem and its applications to vector lattice theory

A classical result in approximation theory states that for any continuous function \( φ: \mathbb{R} \to \mathbb{R} \), the set \( \operatorname{span}\{φ\circ g : g \in \operatorname{Aff}(\mathbb{R})\} \) is dense in \( \mathcal{C}(\mathbb{R}) \) if and only if \( φ\) is not a polynomial. In this note, we present infinite dimensional variants of this result. These extensions apply to neural network architectures and improves 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}.