Whitney's Spectral Synthesis Theorem in Infinite Dimensions

Abstract

This chapter discusses Whitney's spectral synthesis theorem in infinite dimensions. The chapter focuses on extending Whitney's theorem to open subsets of infinite dimensional spaces. In infinite dimensions, there are two equivalent formulations of this theorem. In infinite dimensions, Whitney's theorem is false even in the case U = H, a real separable Hilbert space, and m = l. The chapter presents an example of this. Two other directions arise naturally in infinite dimensions—the first one is to consider subspaces which coincide, in finite dimensions, with the whole space; and the second is to look for a new topology which coincides, in finite dimensions, with the usual one. In addition, the chapter also considers the concept of differentiability type that gives a unified way to deal simultaneously with several subspaces.