Establishing Universal Approximation Theorems (UATs) for nonlinear operators and their derivatives is a foundational open problem in Operator Learning (OL) and raises delicate questions in Nonlinear Functional Analysis. We prove the first UATs for $k$-times differentiable nonlinear operators and their derivatives via OL architectures, uniformly on compact sets and in weighted Bastiani--Sobolev spaces for general finite input measures. In full Banach-space generality, these are the first complete generalizations of the corresponding influential classical UATs in [Hornik, 1991] to infinite-dimensional spaces and OL and they launch Derivative-Informed Operator Learning (DIOL)-learning nonlinear operators and their derivatives-on general Banach spaces. Based on our UATs, we formulate Bastiani--Sobolev training in DIOL. We present open frontiers where DIOL and our UATs find applications: high-order accuracy in OL; fast constrained optimization in Banach spaces (e.g. optimal control of PDEs, inverse problems) via Learn-Then-Optimize; numerical methods for infinite-dimensional PDEs (e.g. HJB PDEs on Banach spaces from infinite-dimensional optimal control via Optimize-Then-Learn, such as optimal control of PDEs, SPDEs, path-dependent systems, partially observed systems, mean-field control). We parameterize nonlinear operators via Encoder-Decoder Architectures, classical OL architectures. These include DeepONets, Deep-H-ONets, and PCA-Nets, which our UATs cover. Our UATs are based on (i) Approximation Properties of Banach spaces; (ii) continuous Bastiani differentiability (weaker than continuous Fréchet differentiability); (iii) $C^k_B$ (Bastiani) compact-open topologies; indeed, UA in $C^k$ (Fréchet) compact-open topologies (induced by operator norms) fails; (iv) construction of weighted Bastiani--Sobolev spaces, generalizing classical Gaussian Sobolev spaces on Banach spaces.

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