Let u and v be vectors in an inner product space V. Prove the Cauchy-Schwarz Inequality for u ≠ 0 as follows:
(a) Let t be a real scalar. Then (tu + v, tu + v) ≥ 0 for all values of t. Expand this inequality to obtain a quadratic inequality of the form
at2 + bt + c ≥ 0
What are a, b, and c in terms of u and v?
(b) Use your knowledge of quadratic equations and their graphs to obtain a condition on a, b, and c for which the inequality in part (a) is true.
(c) Show that, in terms of u and v, your condition in part (b) is equivalent to the Cauchy-Schwarz Inequality.