Let V be a Hilbert space. Let S1and S2 be two hyperplanes in V defined by 

S1={ x ∈V| 〈 a1 , x 〉=b1},   S2={ x ∈V|〈 a2 , x 〉=b2}.

Let y ∈ V be given. We consider the projection of y onto S1∩S2, i.e., the solution of (min x∈S1∩S2) ‖ x − y ‖. 

(a) Prove that S1∩S2 is a plane, i.e., if x ,  z  ∈ S1∩S2, then (1 +t) z −t x ∈ S1∩S2 for any t∈R.

(b) Prove that z is a solution of (1) if and only if z ∈ S1∩S2 and〈 z − y , z − x 〉= 0, ∀ x ∈S1∩S2. 

(c) Find an explicit solution of (1).

(d) Prove the solution found in part (c) is unique.