Some Background Material for 18.176

C(R ;R) is the space of m-times, continuously differentiable funtions, C b (R ;R) is the subspace of C(R ;R) whose elements have bounded derivatives of all orders less than of equal to m, C c (R ;R) is the subspace of C b (R ;R) whose elements have compact support, C 1,2 b ( [0,∞)×R ;R) is the space functions (t,x) ∈ [0,∞) × R 7−→ u(t,x) ∈ R which have one bounded, continuous derivative with respect to t and two bounded, continuous derivatives with respect to x.