Classifiers on relatively compact sets

The problem of classifying signals is of interest in several application areas. Typically we are given a finite number m of pairwise disjoint sets C₁, ..., C_m of signals, and we would like to synthesize a system that maps the elements of each C_j into a real number a_j, such that the numbers a₁, ..., a_m are distinct. In a recent paper it is shown that this classification can be performed by certain simple structures involving linear functionals and memoryless nonlinear elements, assuming that the C_j are compact subsets of a real normed linear space. Here we give a similar solution to the problem under the considerably weaker assumption that the C_j are relatively compact and are of positive distance from each other. An example is given in which the C _j are subsets of L_p(a,b), 1⩽p<∞