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.

翻译：我们利用Leray-Schauder映射，得到了任意Banach空间上连续算子的新通用逼近定理。此外，针对多元函数$L^p$空间中的算子学习问题，我们提出并研究了一种基于多项式基正交投影的方法。在若干附加假设下，我们推导出通过线性投影与有限维映射进行学习的算子通用逼近结果。针对$p=2$的情形，我们给出了保证逼近结果成立的若干充分条件。本文为深度学习方法论建立了理论框架，具体实现将在后续工作中给出。