A note on the existence of continuous functionals

Let if P = {pi|iε{lunate}I} and if Q = {qi|iε{lunate}I} be sets of partial functions with the same index set I. We say that Φ is an interpolating function (from P to Q) if if Φ(pi = qi for each i. We give simple necessary and sufficient conditions for the existence of a monotone interpolating functional. We show that these same conditions are necessary and sufficient for the existence of a continuous interpolating functional if the index set I is finite, but that they are not sufficient if the index set is infinite. © 1981.