The paper establishes the first Universal Approximation Theorems for k‑times differentiable nonlinear operators and their derivatives in general Banach spaces, extending classical results to infinite‑dimensional settings. It introduces Derivative‑Informed Operator Learning (DIOL) and formulates Bastiani–Sobolev training for this framework, covering architectures such as DeepONets, Deep‑H‑ONets, and PCA‑Nets. The work also outlines applications to high‑order accuracy in operator learning, constrained optimization in Banach spaces, and numerical methods for infinite‑dimensional PDEs.
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