柯西–施瓦茨不等式及其证明

一、正向推导

$$
\begin{aligned}
& \ \left| x \right| + \left| y \right| \\ \
= & \ \left(1, 1\right) \cdot \left(\left| x \right|, \left| y \right| \right) \\ \
\leqslant & \ \sqrt{1^{2} + 1^{2}} \sqrt{\left| x \right|^{2} + \left| y \right|^{2}} \\ \
= & \ \sqrt{2} \sqrt{x^{2} + y^{2}} \\ \
\textcolor{lightgreen}{ \leadsto } & \ \textcolor{lightgreen}{ \left| x \right| + \left| y \right| \leqslant \sqrt{2} \sqrt{x^{2} + y^{2}} }
\end{aligned}
$$

上面的 $\textcolor{lightgreen}{ \left| x \right| + \left| y \right| \leqslant \sqrt{2} \sqrt{x^{2} + y^{2}} }$ 这个公式就是柯西–施瓦茨不等式.

二、反向推导

首先：

$$
\begin{aligned}
& \ \left(\left| x \right| + \left| y \right| \right)^{2} = x^{2} + y^{2} + 2 \left| xy \right| \leqslant 2x^{2} + 2y^{2} \\ \
\textcolor{lightgreen}{ \leadsto } & \ \textcolor{gray}{2 \left| xy \right| \leqslant x^{2} + y^{2}} \\ \
\textcolor{lightgreen}{ \leadsto } & \ \left(\left| x \right| + \left| y \right| \right)^{2} \leqslant 2x^{2} + 2y^{2} \\ \
\textcolor{lightgreen}{ \leadsto } & \ \textcolor{gray}{\text{等号两端同时开平方}} \\ \
\textcolor{lightgreen}{ \leadsto } & \ \textcolor{lightgreen}{ \left| x \right| + \left| y \right| \leqslant \sqrt{2} \sqrt{x^{2} + y^{2}} }
\end{aligned}
$$

其中，当且仅当 $\left| x \right| = \left| y \right|$ 时，不等式 $\textcolor{lightgreen}{ \left| x \right| + \left| y \right| \leqslant \sqrt{2} \sqrt{x^{2} + y^{2}} }$ 可以取等号.

三、均值不等式的证明

关于上面“二、反向推导”中用到的 $2 \left| xy \right| \leqslant x^{2} + y^{2}$ 这个均值不等式的证明如下：

$$
\begin{aligned}
& \ \left(\left| x \right| – \left| y \right|\right)^{2} \geqslant 0 \\ \
\textcolor{lightgreen}{ \leadsto } & \ x^{2} – 2 \left| x \right| \left| y \right| + y^{2} \geqslant 0 \\ \
\textcolor{lightgreen}{ \leadsto } & \ \textcolor{gray}{\left|x\right|\left|y\right| = \left|xy\right|} \\ \
\textcolor{lightgreen}{ \leadsto } & \ x^{2}+y^{2}-2\left|xy\right|\geqslant 0 \\ \
\textcolor{lightgreen}{ \leadsto } & \ \textcolor{lightgreen}{2 \left| xy \right| \leqslant x^{2} + y^{2}}
\end{aligned}
$$

其中，当且仅当 $\left| x \right| = \left| y \right|$ 时，不等式 $\textcolor{lightgreen}{2 \left| xy \right| \leqslant x^{2} + y^{2}}$ 可以取等号.

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