# Bochner integrals and neural networks

## TL;DR

This paper develops a functional analytic framework for neural networks using Bochner integrals, establishing new theoretical foundations and properties of variation spaces as Banach spaces.

## Contribution

It introduces a Bochner integral formula for neural networks and analyzes the structure of variation spaces within a functional analytic context.

## Key findings

- Variation spaces are Banach spaces.
- Established norm inequalities relating pointwise and Bochner integrals.
- Derived a Bochner integral formula representing functions via weights and parametrized functions.

## Abstract

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.