Interpolation and Approximation from Convex Sets II. In nite-Dimensional Interpolation

Let X and Y be topological vector spaces, A be a continuous linear map from X to Y , C X, B be a convex set dense in C, and d 2 Y be a data point. Conditions are derived guaranteeing the set B \ A ?1 (d) to be nonempty and dense in C \ A ?1 (d). The paper generalizes earlier results by the authors to the case where Y is innnite dimensional. The theory is illustrated with two examples concerning the existence of smooth monotone extensions of functions deened on a domain of the Euclidean space to a larger domain.