On approximation of linear functionals on Lp spaces

In a recent paper certain approximations to continuous nonlinear functionals defined on an L_p space (1<p<∞) are shown to exist. These approximations may be realized by sigmoidal neural networks employing a linear input layer that implements finite sums of integrals of a certain type. In another recent paper similar approximation results are obtained using elements of a general class of continuous linear functionals. In this note we describe a connection between these results by showing that every continuous linear functional on a compact subset of L_p may be approximated uniformly by certain finite sums of integrals