The Cauchy-Schwarz Inequality - Real Analysis | Lecture 22

The focus for this lecture is one of the most famous and most celebrating inequalities in all of mathematics: the Cauchy-Schwarz inequality. We begin by reminding the viewer of this inequality in Euclidean space, keeping this understanding in a finite-dimensional vector space as motivation for our infinite-dimensional vector space of integrable functions. We then define the scalar product of two Riemann integrable functions which comes as the infinite-dimensional analogue of the dot product in Euclidean space. With the definition of the scalar product, we are then state and prove the Cauchy-Schwarz inequality for Riemann integrable functions.

This course is taught by Jason Bramburger.