Universal Approximation of Continuous Functionals on Compact Subsets via Linear Measurements and Scalar Nonlinearities

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.

💡 Research Summary

The paper establishes a universal approximation theorem for continuous functionals defined on compact subsets of products of Hilbert spaces, and extends the result to Banach‑valued outputs. The authors consider a compact set (K\subset H^{n}) where (H) is a Hilbert space and a continuous map (f:K\to\mathbb R) (or (f:K\to Y) with (Y) a Banach space). They prove that for any tolerance (\varepsilon>0) there exists a representation of the form

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