A Note on the Existence of Continuous Functionals

AbMnwt. Let P = (pi 1 i E I} and 8 = {qi 1 i E I} be sets of partis! functions with the same index set Z. We say that Cp is an interpolating function (from P to 0) if @(pi) = qi for each i. We give simple necessary and sufficient conditions for the existence of a monotone interpolating functional. Wti 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. lc Introduction Recently, there has been interest in languages for functional programming (see [ 1,2]j. In these languages, therl? are two basic entities: (a) partial functions and (b) functionals (which map partial functions into partial functions). Monotone functionals and continuous function& play a fundamental role in this research. We therefore consider basic foundational questions as to their existence. A partial fun&m from A to B is a function from a subset of A to B. In this paper, we will deal with the set SF of Lartial functions from set {O, 1,2,.. . , } of natural numbers. If f is a partial function and/ if a is a nntu& , ' number not in the domain of h then we write f(a) = 1) and1 we say that f(a j is untiefi~ed, or that f is undefined on a. If a and b are elements of N w { I}, th.en we say a E El if either a = b or a = 1. If f and g are partial functions, then we say & g if f(x) G g(x) for all x in hus,