(a) Prove the Cauchy-Schwarz inequality Kylo)|

(a) Prove the CauchySchwarz inequality 11+11. Here we use the shorthand notation (= II Ilk) II) for the norm of Ilk) as on lectures. Hint: Start from 0 II It/J) + A1") II and choose the scalar A ec in a clever way. II 11/1) + 112 and use (a). You may also use the fact that Re(z) Izl for a complex number z. (c) Demonstrate the necessary and sufficient conditions for these inequalities to become equalities. Hint: Let a It/J) It/J) be the projection of l") on to Ilk). (t/JW,') You can write l") in terms of the projection and the rejection I X) = l") as l") = a 11/1) + I X). Note that the rejection is orthogonal to lift), i.e. (xli/J) = 0.