This chapter establishes the exponential decay of gamblets under an appropriate notion of distance derived from subspace decompositionin a way that generalizesdomain decomposition in the computation of PDEs.The first stepspresent sufficient conditions forlocalizationbased on a generalization of the Schwarz subspace decomposition and iterative correction methodintroduced by Kornhuber and Yserentantand the LOD method of Malqvist and Peterseim. However,when equipped withnonconforming measurement functions, one cannot directly work in the primal space, but instead one has to find ways to work in the dual space. Therefore, the next steps presentnecessary and sufficient conditions expressed as frame inequalities in dual spaces that, in applications to linear operators on Sobolev spaces,are expressed as Poincaré, inverse Poincaré, and frame inequalities.