柯西-施瓦茨不等式¶

一般形式¶

设 $n \geq 2$ 是正整数且 $x_{1}, x_{2}, \dots, x_{n}, y_{1}, y_{2}, \dots, y_{n} \in R$

(\sum_{i = 1}^{n} x_{i}^{2}) (\sum_{i = 1}^{n} y_{i}^{2}) \geq {(\sum_{i = 1}^{n} x_{i} y_{i})}^{2}

等号成立当且仅当

\frac{x_{1}}{y_{1}} = \frac{x_{2}}{y_{2}} = \dots = \frac{x_{n}}{y_{n}}

设 $f, g$ 是区间 $[a, b]$ 上的可积函数且平方可积，则有

(\int_{a}^{b} f^{2} (x) d x) (\int_{a}^{b} g^{2} (x) d x) \geq {(\int_{a}^{b} f (x) g (x) d x)}^{2}

⟨ α, α ⟩ ⟨ β, β ⟩ \geq ⟨ α, β ⟩^{2}