  We obtain a new universal approximation theorem for continuous operators on
arbitrary Banach spaces using the Leray-Schauder mapping. Moreover, we
introduce and study a method for operator learning in Banach spaces $L^p$ of
functions with multiple variables, based on orthogonal projections on
polynomial bases. We derive a universal approximation result for operators
where we learn a linear projection and a finite dimensional mapping under some
additional assumptions. For the case of $p=2$, we give some sufficient
conditions for the approximation results to hold. This article serves as the
theoretical framework for a deep learning methodology whose implementation will
be provided in subsequent work.
