Notes on weighted norms and network approximation of functionals

Among other results in the literature concerning arbitrarily good approximation that concern more general types of "target" functionals, different network structures, other nonlinearities, and various measures of approximation errors is the proposition that any continuous real nonlinear functional on a compact subset of a real normed linear space can be approximated arbitrarily well using a single-hidden-layer neural network with a linear functional input layer and exponential (or polynomial or sigmoidal or radial basis function) nonlinearities.