cauchy-schwarz inequality

Cauchy-Schwarz inequality

$\forall \mathbf{x},\mathbf{y}\in {\mathbb{C}}^n$

$$|⟨\mathbf{x},\mathbf{y}⟩|\le \Vert \mathbf{x}{\Vert }_2\cdot \Vert \mathbf{y}{\Vert }_2$$

Furthermore, $|⟨\mathbf{x},\mathbf{y}⟩|=\Vert \mathbf{x}{\Vert }_2\cdot \Vert \mathbf{y}{\Vert }_2⟺\exists \alpha \in \mathbb{C}:\mathbf{x}=\alpha \mathbf{y}$
This inequality is special to the euclidean norm and inner product spaces. It states that the absolute value of the inner product of two vectors is at most the product of their norms, with equality if and only if the vectors are linearly dependent.