Cauchy-Schwarz inequality

The Cauchy-Schwarz inequality is a fundamental concept in mathematics that relates to how closely two vectors (or sequences of numbers) align. It states that the absolute value of their dot product (a measure of their combined magnitude) can never exceed the product of their individual lengths (magnitudes). In simple terms, it provides a maximum limit to how "aligned" two quantities can be, ensuring the dot product isn't disproportionately large compared to their sizes. This inequality is important for understanding angles between vectors and has applications across physics, statistics, and data analysis.