At least, I think it is hard. I don't know how to start this at all.
Let be a vector space equipped with a positive definite inner product. Let be a subspace of . Let be its orthogonal complement. Define a map : -> as where with in and in . Show that:
a) The map is welldefined, linear with range and that . The map is called the projectiion map onto the subspace .
b) Define . Show that is the projection map onto i.e. show that is a linear map with range and that .