The Cauchy-Schwarz inequality : Proofs and applications in various spaces

We give some background information about the Cauchy-Schwarz inequality including its history. We then continue by providing a number of proofs for the inequality in its classical form using various proof techniques, including proofs without words. Next we build up the theory of inner product spaces from metric and normed spaces and show applications of the Cauchy-Schwarz inequality in each content, including the triangle inequality, Minkowski's inequality and Holder's inequality. In the final part we present a few problems with solutions, some proved by the author and some by others.