We study universal approximation of continuous functionals on compact subsets of products of Hilbert spaces. We prove that any such functional can be uniformly approximated by models that first take finitely many continuous linear measurements of the inputs and then combine these measurements through continuous scalar nonlinearities. We also extend the approximation principle to maps with values in a Banach space, yielding finite-rank approximations. These results provide a compact-set justification for the common ``measure, apply scalar nonlinearities, then combine'' design pattern used in operator learning and imaging.

翻译：本文研究希尔伯特空间积空间紧子集上连续泛函的通用逼近问题。我们证明，任意此类泛函均可通过以下模型实现一致逼近：首先对输入进行有限个连续线性测量，再通过连续标量非线性变换组合这些测量结果。我们进一步将该逼近原理推广至取值于巴拿赫空间的映射，从而得到有限秩逼近。这些结果为算子学习与成像领域中常用的“测量-施加标量非线性变换-组合”设计模式提供了紧集层面的理论依据。