A universal reproducing kernel Hilbert space for learning nonlinear systems operators

The paper proves density and completeness for a product RKHS of operators by importing an RBF universal-approximation theorem, then showing the RBF approximant is an RKHS interpolant and invoking best-approximation in the RKHS (see k⊗, K⊗, and interpolant identities, as well as the main Theorem 2 statements and proof steps) . The candidate solution instead gives a direct RKHS proof via density of kernel sections, projection, and reproducing bounds; it also notes K⊗ = Ku ⊗ Kx and projection=interpolant when the Gram is invertible, and constructs a complete ON system by Gram–Schmidt—arriving at the same conclusions through a different route. The paper’s proof has a small gap where it moves from data-dependent RBF scales to fixed (data-independent) kernel hyperparameters; this can be patched by standard universality results for fixed-bandwidth radial kernels, but it should be stated or cited explicitly. Overall, both are correct; the proofs are different.