12.4. Coefficients relative to a smooth frame

Let : E M be a C vector bundle and s1,...,sr a C frame for E over an open set U in M. Then every e ^1(U) can be written uniquely as a linear combination

e= (j=1to r) cj(e)sj(p), p=(e)U.

Prove that cj : 1U R is C for j = 1,...,r. (Hint: First show that the coefficients of erelative to the frame t1,...,tr of a trivialization are C.)