Projection of a function onto the orthogonal complement of a subspace

Question

I have a polynomial subspace, lets say $U$, which I applied the Gram-Schmidt algorithm to find an orthonormal basis. I had to find the projection of $cosh(x)$ onto $U$, so as I had found the orthonormal basis, I just used the formula $$Proj_Ucosh(x) = \lt cosh(x), U_1\gt U_1 + ... + \lt cosh(x), U_k\gt U_k$$

Is there a similar formula to find the projection of $cosh(x)$ onto $U^{\perp}$?

Answer 1

Hint: $ proj_U f + proj_{U^\perp} f = f $.