  Reproducing Kernel Hilbert spaces (RKHS) have been a very successful tool in
various areas of machine learning. Recently, Barron spaces have been used to
prove bounds on the generalisation error for neural networks. Unfortunately,
Barron spaces cannot be understood in terms of RKHS due to the strong nonlinear
coupling of the weights. This can be solved by using the more general
Reproducing Kernel Banach spaces (RKBS). We show that these Barron spaces
belong to a class of integral RKBS. This class can also be understood as an
infinite union of RKHS spaces. Furthermore, we show that the dual space of such
RKBSs, is again an RKBS where the roles of the data and parameters are
interchanged, forming an adjoint pair of RKBSs including a reproducing kernel.
This allows us to construct the saddle point problem for neural networks, which
can be used in the whole field of primal-dual optimisation.
