Suppose that we have an inner product space

and a finite dimensional subspace

.

Let

. Prove that

for all

and

in

.

I've gone about various ways of trying to prove this; my main attack of this problem was to write out a basis for W, and then expand out the projection expression in terms of each basis vector, and then write out the inner products and see what I get (direct proof). This just leads me to a dead end.

I was able to sketch a quick proof in the Euclidean space with the dot product, but of course this is much too restrictive to be a proof (it's easier when you know exactly how the dot product is defined). This led me to the idea of representing the arbitrary dot product as a matrix, and then setting up the equations and trying another direct proof, but again it looks messy, and it seems like this kind of direct proof could get extremely long and bogged down, especially when you start expanding out individual linear combinations and distributing across inner products, etc. etc.

I need a new idea to approach this problem...any help would be appreciated! =)