We develop a functional framework for shallow neural networks based on reproducing kernel Banach spaces. This approach enables a nonparametric treatment of neural networks, in direct analogy with kernel methods. A representer theorem shows that finite networks suffice for empirical risk minimization. Estimation and approximation error bounds can then be derived in linear function spaces. As a byproduct, we obtain universality results and approximation bounds showing that neural networks can adapt to latent structure in the problem. Further, we derive complexity estimates based on the Rademacher complexities of RKBS balls, independent of network size.