Vector-Valued RKBS Framework for Neural Networks and Operators

A new mathematical framework extends reproducing kernel Banach spaces (RKBS) to vector-valued functions, covering neural networks and neural operators. The authors develop adjoint pairs of vector-valued RKBS (vv-RKBS) with an associated kernel, proving every vv-RKBS belongs to such a pair. This construction avoids restrictive assumptions like symmetric domains, finite-dimensional outputs, reflexivity, or separability, while retaining properties of vector-valued RKHS. They show shallow ℝ^d-valued neural networks are elements of a specific vv-RKBS. The work addresses a gap in understanding function spaces for multi-output networks and operator models.