Bochner integrals and neural networks

A Bochner integral formula is derived that represents a function in terms of weights and a parametrized family of functions. Comparison is made to pointwise formulations, norm inequalities relating pointwise and Bochner integrals are established, variation-spaces and tensor products are studied, and examples are presented. The paper develops a functional analytic theory of neural networks and shows that variation spaces are Banach spaces.