Neural Networks in Fr\'echet spaces

We derive approximation results for continuous functions from a Fr\'echet space $\mathfrak X$ into its field $\mathbb{F}$. The approximation is similar to the well known universal approximation theorems for continuous functions from $\mathbb{R}^n$ to $\mathbb{R}$, where approximation is done with (multilayer) neural networks [10, 16, 12, 20]. Similar to classical neural networks, the approximating functions that we obtain are easy to implement and allows for fast computation and fitting.