Theorem: (Cauchy-Schwarz Inequality)
令 $V$ 為實數內積空間，且 ${\bf u}, {\bf v} \in V$ 則\[
|({\bf u}, {\bf v})| \le ||{\bf u}|| \; ||{\bf v}||
\]==============================
先看幾個例子
Example 1: 歐幾里德平面空間對應的 柯西不等式
$V:=\mathbb{R}^2$ 且配備標準內積 $({\bf u}, {\bf v}) := {\bf u}^T {\bf v}$，現令 ${\bf u}:=[u_1\;\;u_2]^T; \; {\bf v}:=[v_1\;\;v_2]^T$ 則上述的 Cauchy-Schwarz Inequality 可表為
\[\begin{array}{l}
\left| {\left( {{\bf{u}},{\bf{v}}} \right)} \right| \le \left\| {\bf{u}} \right\|\left\| {\bf{v}} \right\|\\
\Rightarrow {\left| {{{\bf{u}}^T}{\bf{v}}} \right|^2} \le \left( {{{\bf{u}}^T}{\bf{u}}} \right)\left( {{{\bf{v}}^T}{\bf{v}}} \right)\\
\Rightarrow {\left| {{u_1}{v_1} + {u_2}{v_2}} \right|^2} \le \left( {u_1^2 + u_2^2} \right)\left( {v_1^2 + v_2^2} \right)
\end{array}\]
Example 2: 有限維歐幾里德空間對應的 柯西不等式
$V:=\mathbb{R}^n$ 且配備標準內積 $({\bf u}, {\bf v}) := {\bf u}^T {\bf v}$，現令 ${\bf u}:=[u_1,...,\;\;u_n]^T; \; {\bf v}:=[v_1,...,\;\;v_n]^T$ 則上述的 Cauchy-Schwarz Inequality 可表為
\[\begin{array}{*{20}{l}}
{\left| {\left( {{\bf{u}},{\bf{v}}} \right)} \right| \le \left\| {\bf{u}} \right\|\left\| {\bf{v}} \right\|}\\
{ \Rightarrow {{\left| {{{\bf{u}}^T}{\bf{v}}} \right|}^2} \le \left( {{{\bf{u}}^T}{\bf{u}}} \right)\left( {{{\bf{v}}^T}{\bf{v}}} \right)}\\
{ \Rightarrow {{\left| {{u_1}{v_1} + {u_2}{v_2} + ... + {u_n}{v_n}} \right|}^2} \le \left( {u_1^2 + u_2^2 + ... + u_n^2} \right)\left( {v_1^2 + v_2^2 + ... + v_n^n} \right)}
\end{array}\]
Example 3: 無窮維 實數連續函數空間 對應的 柯西不等式
$V:=C[0,1]$ 且配備內積 $(f(t), g(t)) := \int_0^1 f(t) g(t) dt$，現令 ${\bf u}:=f(t); \; {\bf v}:=g(t)$ 則上述的 Cauchy-Schwarz Inequality 可表為
\[\begin{array}{l}
\left| {\left( {f\left( t \right),g\left( t \right)} \right)} \right| \le \left\| {f\left( t \right)} \right\|\left\| {g\left( t \right)} \right\|\\
\Rightarrow {\left| {\int_0^1 {f\left( t \right)g\left( t \right)dt} } \right|^2} \le \left( {\int_0^1 {{f^2}\left( t \right)dt} } \right)\left( {\int_0^1 {{g^2}\left( t \right)dt} } \right)
\end{array}\]
Example 4: 實數矩陣空間對應的柯西不等式
令 $V:= M_{n \times n}$ 且配備內積 $(A, B) := tr(B^T A)$ 現令 ${\bf u}:=A; \; {\bf v}:=B$ 為 $n \times n$ 矩陣，則上述的 Cauchy-Schwarz Inequality 可表為
\[\begin{array}{*{20}{l}}
{\left| {\left( {{\bf{u}},{\bf{v}}} \right)} \right| \le \left\| {\bf{u}} \right\|\left\| {\bf{v}} \right\|}\\
{ \Rightarrow {{\left| {tr\left( {{B^T}A} \right)} \right|}^2} \le tr\left( {{A^T}A} \right)tr\left( {{B^T}B} \right)}
\end{array}\]
Example 5: 隨機變數所成的 $L^2$ 空間之柯西不等式：
令 $V:= L^p :=\{X: E[|X|^2] < \infty\}$ 且配備內積 $(X,Y) := E[XY]$，其中 $X,Y$ 為隨機變數，$E[\cdot]$ 表期望值。現令 ${\bf u} := X$ 且 ${\bf v} := Y$ 則上述的 Cauchy-Schwarz Inequality 可表為
\begin{align*}
& \left| {\left( {{\mathbf{u}},{\mathbf{v}}} \right)} \right| \leq \left\| {\mathbf{u}} \right\|\left\| {\mathbf{v}} \right\| \hfill \\
& \Rightarrow \left| {E\left[ {XY} \right]} \right| \leq \left\| X \right\|\left\| Y \right\| \hfill \\
& \Rightarrow \left| {E\left[ {XY} \right]} \right| \leq \sqrt {E\left[ {{X^2}} \right]} \sqrt {E\left[ {{Y^2}} \right]} \hfill \\
\end{align*}
Comments:
1.上述幾個例子展示了儘管所表現的樣式非常不同，但從抽象化觀點而言是同一件事情。
2. 柯西等式何時成立？
以下我們給出 Cauchy-Schwarz Inequality 的證明，此證明頗具巧思有興趣的讀者可細細品味。
Proof of Cauchy-Schwarz Inequality
令 $c \in \mathbb{R}^1$ 現在觀察
\[\begin{array}{l}
\underbrace {\left( {{\bf{u}} - c{\bf{v}},{\bf{u}} - c{\bf{v}}} \right)}_{ = {{\left\| {{\bf{u}} - c{\bf{v}}} \right\|}^2}} = \left( {{\bf{u}},{\bf{u}}} \right) + \left( { - c{\bf{v}},{\bf{u}}} \right) + \left( {{\bf{u}}, - c{\bf{v}}} \right) + \left( { - c{\bf{v}}, - c{\bf{v}}} \right)\\
= \left( {{\bf{u}},{\bf{u}}} \right) - c\left( {{\bf{v}},{\bf{u}}} \right) - c\left( {{\bf{u}},{\bf{v}}} \right) + {c^2}\left( {{\bf{v}},{\bf{v}}} \right)\\
= \left( {{\bf{u}},{\bf{u}}} \right) - 2c\left( {{\bf{u}},{\bf{v}}} \right) + {c^2}\left( {{\bf{v}},{\bf{v}}} \right) \;\;\;\; (*)
\end{array}\]
若 ${\bf{v}} \ne 0$ 則 $({\bf v},{\bf v})>0$ 我們可取 \[c: = \frac{{\left( {{\bf{u}},{\bf{v}}} \right)}}{{\left( {{\bf{v}},{\bf{v}}} \right)}}\]將此 $c$ 帶入 $(*)$ 可得
\[\begin{array}{l}
{\left\| {{\bf{u}} - c{\bf{v}}} \right\|^2} = \left( {{\bf{u}},{\bf{u}}} \right) - 2c\left( {{\bf{u}},{\bf{v}}} \right) + {c^2}\left( {{\bf{v}},{\bf{v}}} \right)\\
= \left( {{\bf{u}},{\bf{u}}} \right) - 2\frac{{\left( {{\bf{u}},{\bf{v}}} \right)}}{{\left( {{\bf{v}},{\bf{v}}} \right)}}\left( {{\bf{u}},{\bf{v}}} \right) + {\left( {\frac{{\left( {{\bf{u}},{\bf{v}}} \right)}}{{\left( {{\bf{v}},{\bf{v}}} \right)}}} \right)^2}\left( {{\bf{v}},{\bf{v}}} \right)\\
= \left( {{\bf{u}},{\bf{u}}} \right) - \frac{{{{\left( {{\bf{u}},{\bf{v}}} \right)}^2}}}{{\left( {{\bf{v}},{\bf{v}}} \right)}}
\end{array}\]但由於 ${\left\| {{\bf{u}} - c{\bf{v}}} \right\|^2} \ge 0$ 故
\[\left( {{\bf{u}},{\bf{u}}} \right) - \frac{{{{\left( {{\bf{u}},{\bf{v}}} \right)}^2}}}{{\left( {{\bf{v}},{\bf{v}}} \right)}} \ge 0 \Rightarrow \left( {{\bf{u}},{\bf{u}}} \right)\left( {{\bf{v}},{\bf{v}}} \right) \ge {\left( {{\bf{u}},{\bf{v}}} \right)^2}\]
上式結果說明在 ${\bf v} \neq 0$ 時 Cauchy-Schwarz Inequality 成立。另外我們回頭檢驗 ${\bf v} = 0$ 的情況，則此時 $\left( {{\bf{u}},{\bf{u}}} \right)\left( {{\bf{v}},{\bf{v}}} \right) \ge {\left( {{\bf{u}},{\bf{v}}} \right)^2}$ 自動滿足。故不論如何我們都有
\[{\left( {{\bf{u}},{\bf{v}}} \right)^2} \le \left( {{\bf{u}},{\bf{u}}} \right)\left( {{\bf{v}},{\bf{v}}} \right)\]至此證畢。$\square$
Comments:
1. 注意到若 ${\bf u} = c {\bf v} $ 則 Cauchy-Schwarz 等式成立。此結果背後蘊含最小平方的最佳化觀點但我們在此不作贅述。
2. 上述 Cauchy-Schwarz Inequality 可引出 Triangular Inequality
Corollary: Triangular Inequality
令 $V$ 為實數內積空間，若 ${\bf u}, {\bf v} \in V $ 則
\[
||{\bf u} + {\bf v}|| \le ||{\bf u}|| + ||{\bf v}||
\]
Proof:
觀察
\[\begin{array}{l}
||{\bf{u}} + {\bf{v}}|{|^2} = \left( {{\bf{u}} + {\bf{v}},{\bf{u}} + {\bf{v}}} \right)\\
\begin{array}{*{20}{c}}
{}&{}&{}
\end{array} = \left( {{\bf{u}},{\bf{u}}} \right) + \left( {{\bf{v}},{\bf{u}}} \right) + \left( {{\bf{u}},{\bf{v}}} \right) + \left( {{\bf{v}},{\bf{v}}} \right)\\
\begin{array}{*{20}{c}}
{}&{}&{}
\end{array} = \left( {{\bf{u}},{\bf{u}}} \right) + 2\left( {{\bf{v}},{\bf{u}}} \right) + \left( {{\bf{v}},{\bf{v}}} \right)\\
\begin{array}{*{20}{c}}
{}&{}&{}
\end{array} = {\left\| {\bf{u}} \right\|^2} + 2\left( {{\bf{v}},{\bf{u}}} \right) + {\left\| {\bf{v}} \right\|^2}\\
\begin{array}{*{20}{c}}
{}&{}&{}
\end{array} \le {\left\| {\bf{u}} \right\|^2} + 2\left| {\left( {{\bf{v}},{\bf{u}}} \right)} \right| + {\left\| {\bf{v}} \right\|^2}
\end{array}\]由 Cauchy-Schwarz Inequality 我們有 $
|({\bf u}, {\bf v})| \le ||{\bf u}|| \; ||{\bf v}||$ 故
\[\begin{array}{l}
||{\bf{u}} + {\bf{v}}|{|^2} \le {\left\| {\bf{u}} \right\|^2} + 2\left| {\left( {{\bf{v}},{\bf{u}}} \right)} \right| + {\left\| {\bf{v}} \right\|^2}\\
\begin{array}{*{20}{c}}
{}&{}&{}
\end{array} \le {\left\| {\bf{u}} \right\|^2} + 2||{\bf{u}}||\;||{\bf{v}}|| + {\left\| {\bf{v}} \right\|^2}\\
\begin{array}{*{20}{c}}
{}&{}&{}
\end{array} \le {\left( {\left\| {\bf{u}} \right\| + \left\| {\bf{v}} \right\|} \right)^2}
\end{array}\]對兩邊同開根號我們得到
\[
||{\bf u} + {\bf v}|| \le ||{\bf u}|| + ||{\bf v}||
\]即為所求 $\square$
Comment:
若 ${\bf u}, {\bf v}$ 互為正交，亦即 $({\bf u},{\bf v}) = 0$ 則我們有
\[
||{\bf u} + {\bf v}||^2 = ||{\bf u}||^2 + ||{\bf v}||^2
\]上述等式 可視為 畢氏定理 在向量空間中的推廣。
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