  We study two-layer neural networks whose domain and range are Banach spaces
with separable preduals. In addition, we assume that the image space is
equipped with a partial order, i.e. it is a Riesz space. As the nonlinearity we
choose the lattice operation of taking the positive part; in case of $\mathbb
R^d$-valued neural networks this corresponds to the ReLU activation function.
We prove inverse and direct approximation theorems with Monte-Carlo rates for a
certain class of functions, extending existing results for the
finite-dimensional case. In the second part of the paper, we study, from the
regularisation theory viewpoint, the problem of finding optimal representations
of such functions via signed measures on a latent space from a finite number of
noisy observations. We discuss regularity conditions known as source conditions
and obtain convergence rates in a Bregman distance for the representing measure
in the regime when both the noise level goes to zero and the number of samples
goes to infinity at appropriate rates.
