Cauchy-Schwarz Inequality: Proof and Applications

This lecture covers the Cauchy-Schwarz inequality, proving that for a real inner product space, the square of the inner product of two vectors is less than or equal to the product of their norms. It also explores the relationship between norms and inner products, demonstrating that the norm induced by an inner product satisfies the Cauchy-Schwarz inequality. Additionally, it discusses the equivalence of norms on a real inner product space and the Euclidean norm. The lecture concludes with the application of the Cauchy-Schwarz inequality in various contexts.