  Scale structures were introduced by H. Hofer, K. Wysocki, and E. Zehnder as a
new concept of a smooth structure in infinite dimensions. We prove that scale
structures on mapping spaces are completely determined by the dimension of
domain manifolds. As a consequence, we give a complete description of the local
invariant introduced by U. Frauenfelder for mapping spaces. Product mapping
spaces and relative mapping spaces are also studied. Our approach is based on
the spectral resolution of Laplace type operators together with the eigenvalue
growth estimate.
