ON APPROXIMATION OF LINEAR FUNCTIONALS ON Lp SPACES

In a recent paper certain approximations to continuous nonlinear functionals de-ned on an L p space (1 < p < 1) are shown to exist. These approximations may be realized by sigmoidal neural networks employing a linear input layer that implements nite 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 nite sums of integrals.