Show that L2(,F, P) is a Hilbert space with inner product

X, Y = E[XY] and if E is a sub--algebra of F, then L2(, E, P) is a

subspace of L2(,F, P). Show that if X is a random variable in L2(,F, P),

then E[X|E] is the orthogonal projection of X onto the subspace L2(, E, P).