1. * (a) Prove the Schwarz inequality ||a|| ||B|| | (a|)|. Hint: use the fact that ||a + c|| 0 for any c E C. (b) Prove that the equality holds only if |B) = dla), with c E C. (c) Prove the triangular inequalities ||||||||||||a + B|| ||a|| + || B||. (d) Consider a) and 3) in a discrete Hilbert space. Prove the Schwarz inequality by considering the expan of a) and 3) in a basis {01), 2),...}.