Cauchy-Schwartz Inequality

For any $$x,y \in \mathbb{R}^n$$, it holds that $$|x^T y| \le ||x||\,||y||$$.

An easy way to remember this is to relate it to the angle between two vectors. For the angle $$\theta = \angle(x,y) = \cos^{-1} \frac{x^T y}{||x||\,||y||}$$ to be real, the argument of the $$\cos^{-1}$$ at the right should be in the range $$[-1,1]$$. This is only possible if $$- ||x||\,||y|| \le |x^T y| \le ||x||\,||y||$$.