Thus, Π Π assigns to each point x a linear map Π Π x , which is a linear map from T into the space of distributions on R d ; Γ Γ assigns to any two points x and y a bounded operator Γ Γ x y , which has the role of converting an expansion based at y into one based at x . These maps Π Π and Γ Γ are required to satisfy the algebraic conditions