---
tag: WCN2JWVTRB
kind: statement
role: proposition
name: Algebraic laws
sources:
  - wikipedia: en:77662984@r1362840840
    locator: Law (mathematics), section Notable examples > Algebraic laws
    note: adopted
    mention: wikipedia://en/77662984#NCYG
    sections: [Statement]
inventory: [23PVH9E0VJ]
---

## Statement

[[Cauchy–Schwarz inequality|0RFWW56Y22]]: An upper bound on the [[inner product|01EK923SKP]] between two [[vectors|0QN3D80AWP]] in an inner product space in terms of the product of the vector [[norms|0V2WP9B1MF]]. It is considered one of the most important and widely used inequalities in mathematics.

The Cauchy–Schwarz inequality states that for all vectors $\mathbf{u}$ and $\mathbf{v}$ of an [[inner product space|01EK923SKP]]

$$\left \vert \langle \mathbf u, \mathbf v \rangle \right \vert \leq \langle \mathbf u, \mathbf u \rangle \cdot \langle \mathbf v, \mathbf v \rangle$$

where $\langle \cdot, \cdot \rangle$ is the inner product. Examples of inner products include the real and complex [[dot product|0S52TNNTH9]]; see the [[examples in inner product|01EK923SKP]]. Every inner product gives rise to a Euclidean $l_2$ [[norm|0V2WP9B1MF]], called the canonical or [[induced norm|01EK923SKP]], where the norm of a vector $\mathbf{u}$ is denoted and defined by

$$\|\mathbf{u}\| := \sqrt{\langle \mathbf{u}, \mathbf{u} \rangle},$$

where $\langle \mathbf{u}, \mathbf{u} \rangle$ is always a non-negative real number (even if the inner product is complex-valued). By taking the square root of both sides of the above inequality, the Cauchy–Schwarz inequality can be written in its more familiar form in terms of the norm:

$$\left \vert \langle \mathbf u, \mathbf v \rangle \right \vert \leq \langle \mathbf u, \mathbf u \rangle \cdot \langle \mathbf v, \mathbf v \rangle$$

Moreover, the two sides are equal if and only if $\mathbf{u}$ and $\mathbf{v}$ are [[linearly dependent|0CRMGPAH83]].