On finite-dimensional encoding/decoding theorems for neural operators

Recently, versions of neural networks with infinite-dimensional affine operators inside the computational units (``neural operator'' networks) have been applied to learn solutions to differential equations. To enable practical computations, one employs finite-dimensional encoding/decoding theorems of the following kind: every continuous mapping $f$ between function spaces $E$ and $F$ is approximated in the topology of uniform convergence on compacta by continuous mappings factoring through two finite dimensional Banach spaces. Such a result is known (Kovachki et al., 2023) for $E,F$ being Banach spaces having the approximation property. We point out that the result needs no assumptions on $E,F$ whatsoever and remains true not only for all normed spaces, but for arbitrary locally convex spaces as well. At the same time, an analogous result for $C^k$-smooth mappings and the $C^k$ compact open topology, $k\geq 1$, holds if and only if the space $E$ has the approximation property. This analysis may be useful already because non-normable locally convex function spaces are common in the theory of differential equations, the main field of applications for the emerging theory.