Today I want to present a deceptively simple fact about spaces equipped with inner products. The Cauchy-Schwarz inequality states that
for any vectors . The proof uses a neat little trick. We take a scalar and construct the vector . Now the positive-definiteness, bilinearity, and symmetry of the inner product tells us that
This is a quadratic function of the real variable . It can have at most one zero, if there is some value such that is the zero vector, but it definitely can’t have two zeroes. That is, it’s either a perfect square or an irreducible quadratic. Thus we consider the discriminant and conclude
which is easily seen to be equivalent to the Cauchy-Schwarz inequality above. As a side effect, we see that we only get an equality (rather than an inequality) when and are linearly dependent.