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

S1={xV| a1,x=b1}, S2={xV|a2,x=b2}.

Lety V be given. We consider the projection ofyonto S1S2, i.e., the solution of (min xS1S2) xy.

(a) Prove that S1S2 is a plane, i.e., ifx,z S1S2, then (1 +t)ztx S1S2 for any tR.

(b) Prove thatzis a solution of (1) if and only ifz S1S2 andzy,zx= 0, xS1S2.

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

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