# 柯西-施瓦茨不等式

## 2. 表述

$\begin{array}{lll} |\langle \mathbf{u}, \mathbf{v} \rangle|^2 \leq \langle \mathbf{u},\mathbf{u} \rangle \cdot \langle \mathbf{v},\mathbf{v} \rangle \end{array}$

$\begin{array}{lll} & = & t\langle \mathbf{u},t\mathbf{u}+c\mathbf{v} \rangle + c\langle \mathbf{v},t\mathbf{u}+c\mathbf{v} \rangle \\ & = & \langle \mathbf{u},\mathbf{u} \rangle t^2 + c^*\langle \mathbf{u},\mathbf{v} \rangle t + c\langle \mathbf{v},\mathbf{u} \rangle t + cc^*\langle \mathbf{v},\mathbf{v} \rangle \\ & = & \langle \mathbf{u},\mathbf{u} \rangle t^2 + 2|\langle \mathbf{u},\mathbf{v} \rangle|^2 t + |\langle \mathbf{u},\mathbf{v} \rangle|^2 \langle \mathbf{v},\mathbf{v} \rangle \\ & \geq & 0 \end{array}$

$\begin{array}{llll} &\Delta & = & (2|\langle \mathbf{u},\mathbf{v} \rangle|^2)^2 - 4\langle \mathbf{u},\mathbf{u} \rangle |\langle \mathbf{u},\mathbf{v} \rangle|^2 \langle \mathbf{v},\mathbf{v} \rangle \\ & & \leq & 0 \\ \Rightarrow & |\langle \mathbf{u},\mathbf{v} \rangle|^2 & \leq & \langle \mathbf{u},\mathbf{u} \rangle \langle \mathbf{v},\mathbf{v} \rangle \end{array}$

Q.E.D.

## 3. 特例

### 3.1 $\mathbb{R}^2$ 空间

$\begin{array}{lll} (u_1 v_1 + u_2 v_2)^2 \leq (u_1^2 + u_2^2)(v_1^2 + v_2^2) \end{array}$

### 3.2 $\mathbb{R}^n$ 空间

$\begin{array}{lll} (\sum_{i=1}^n u_i v_i)^2 \leq (\sum_{i=1}^n u_i^2)(\sum_{i=1}^n v_i^2) \end{array}$

### 3.3 $\mathbb{C}^n$ 空间

$\begin{array}{lll} |u_1 \bar{v}_1 + \cdots + u_n \bar{v}_n |^2 \leq (|u_1|^2 + \cdots + |u_n|^2)(|v_1|^2 + \cdots + |v_n|^2) \end{array}$

### 3.4 $\mathbf{L}^2$ 空间

$\mathbf{L}^2$ 空间是定义在均方可积下的复函数内积空间，对应的 CS 不等式为：

$\begin{array}{lll} |\int_{\mathbb{R}^n} f(x)\overline{g(x)} dx |^2 \leq \int_{\mathbb{R}^n} |f(x)|^2 dx \int_{\mathbb{R}^n} |g(x)|^2 dx \end{array}$