Question 1. [6, 2, 2 marks] (a) Prove that if (,) and (,)2 are two inner products on a real inner product space H with associated norms ||· ||₁ and ||·||2 satisfying . ||f||1 = ||f||2 for all fe H, then (f,g)1 = (f,g)2 for all f, g € H. (HINT: First prove the Polarization identity (f,g) = (f + g||² + ||ƒ − g||²)). (b) Give a geometrical interpretation of (a). (c) For complex inner product space, prove that 4Re(f, g) = (||ƒ+g||²+||ƒ-g||²).

Fig: 1