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Various examples of the inequality of Cauchy-Schwarz.

the space the inner product the inequality $\begin{vmatrix}\langle \overrightarrow{x} ,\overrightarrow{y}\rangle
\end{vma...
...ightarrow{x}\end{Vmatrix} \cdot
\begin{Vmatrix}\overrightarrow{y}\end{Vmatrix}$
$\mathbb{R} ^2 $ $\langle
\begin{pmatrix}x_1 \\ x_2 \end{pmatrix}, \begin{pmatrix}y_1 \\ y_2 \end{pmatrix}\rangle = x_1y_1+x_2y_2$ $ \begin{vmatrix}x_1y_1+x_2y_2 \end{vmatrix} \leqslant
\left( x_1^2+x_2^2 \right)^{\frac 12} \cdot
\left( y_1^2+y_2^2 \right)^{\frac 12}$
$\mathbb{R} ^2 $ $\langle
\begin{pmatrix}x_1 \\ x_2 \end{pmatrix}, \begin{pmatrix}y_1 \\ y_2 \end{pmatrix}\rangle = \lambda_1 x_1y_1+ \lambda_2 x_2y_2$ $\begin{vmatrix}\lambda_1 x_1y_1+\lambda_2 x_2y_2 \end{vmatrix} \leqslant
\left...
...t)^{\frac 12} \cdot
\left( \lambda_1 y_1^2+ \lambda_2 y_2^2 \right)^{\frac 12}$
  $\lambda_1,\lambda_2 > 0$  
$\mathbb{R} ^n$ $\left\langle
\begin{pmatrix}x_1 \\ \dots \\ x_n \end{pmatrix},
\begin{pmatrix...
...s \\ y_n \end{pmatrix}\right\rangle =
\underset{i=1}{\overset{n}{\sum}} x_i y_i$ $\underset{i=1}{\overset{n}{\sum}} x_i y_i \leqslant
\left( \underset{i=1}{\over...
...rac 12} \cdot
\left( \underset{i=1}{\overset{n}{\sum}} y_i^2 \right)^{\frac 12}$
$\mathbb{R} ^n$ $\langle
\begin{pmatrix}x_1 \\ \dots \\ x_n \end{pmatrix},
\begin{pmatrix}y_1 ...
...\ y_n \end{pmatrix}\rangle =
\underset{i=1}{\overset{n}{\sum}}\lambda_i x_i y_i$ $ \begin{vmatrix}\underset{i=1}{\overset{n}{\sum}}\lambda_i x_i y_i
\end{vmatr...
...cdot
\left( \underset{i=1}{\overset{n}{\sum}}\lambda_i y_i^2 \right)^{\frac 12}$
  $\lambda_i \geq 0, \forall i$  
C[a,b] $\langle f, g \rangle = \int_a^b f(x)g(x)dx$ $\begin{vmatrix}\int_a^b f(x)g(x)dx \end{vmatrix}
\leqslant \left( \int_a^b f(x)^2dx \right)^{\frac 12}
\cdot \left( \int_a^b g(x)^2dx \right)^{\frac 12}$
C[a,b] $\langle f, g \rangle = \int_a^b \lambda(x)f(x)g(x)dx$ $ \begin{vmatrix}\int_a^b \lambda(x)f(x)g(x)dx \end{vmatrix} \leqslant
\left( \...
...\right)^{\frac 12} \cdot
\left( \int_a^b \lambda(x)g(x)^2dx \right)^{\frac 12}$
  $\lambda(x) > 0 , \forall x$  

Definition 15.0.1   In the last line, the positive function $\lambda$ is called A weight, and the corresponding inner product is called a weighted integral inner product.

Example 15.0.2   Examples of inequalities which are consequences of Cauchy-Schwarz inequality:


$V=\mathbb{R} ^2:$

$\forall x_1,x_2,y_1,y_2 \in \mathbb{R} $: $(x_1y_1+x_2y_2)^2 \leqslant (x_1^2+x_2^2)\cdot (y_1^2+y_2^2)$
y1=y2=1 $\Rightarrow$ $\underbrace{(\overset{5}{x_1} +\overset{7}{x_2} )^2}_{144}
\leqslant \underbrace{2(x_1^2+x_2^2)}_{148}$
     
y1=1, y2=3 $\Rightarrow$ $ (x_1+3x_2)^2 \leqslant 10(x_1^2+x_2^2)$
     
y1=1, y2=-1 $\Rightarrow$ $ (x_1-x_2)^2 \leqslant 2(x_1^2+x_2^2)$

$\quad$





V=C[0,1]:

$\forall f,g \in C[0,1]$: $\int_0^1 f(x)g(x)dx \leqslant \left( \int_0^1 f(x)^2 dx \right)^{\frac 12} \cdot
\left( \int_0^1 g(x)^2 dx \right)^{\frac 12}$ $\int_0^1 e^x \sin x dx \leqslant \left( \int_0^1 e^{2x} dx \right)^{\frac 12} \cdot
\left( \int_0^1 \sin^2 x dx \right)^{\frac 12}$ or $\int_0^1 e^x \sin x dx \leqslant
\sqrt{\frac12 \left( e^2-1 \right) \left(1-\frac{\sin 2}{2} \right) }$




   
   


$V=\mathbb{R} ^3:$ $\lambda_1=3, \lambda_2=2, \lambda_3=1$.

$\forall x_1,x_2,x_3,y_1,y_2,y_3 \in \mathbb{R} $: $(3x_1y_1+2x_2y_2+x_3y_3)^2 \leqslant
(3x_1^2+2x_2^2+x_3^2)\cdot (3y_1^2+2y_2^2+y_3^2)$ y1=y2=y3=1

$\Downarrow$

$\underbrace{(3\overset{1}{x_1} +2\overset{3}{x_2} +\overset{5}{x_3} )}_{196}
\leqslant \underbrace{6(3x_1^2+2x_2^2+x_3^2)}_{276}$






V=C[1,2]

$\forall f,g \in C[1,2]$: $\int_1^2 f(x)g(x)dx \leqslant \left( \int_1^2 f(x)^2 dx \right)^{\frac 12} \cdot
\left( \int_1^2 g(x)^2 dx \right)^{\frac 12}$ $\int_1^2 \frac{\sin x}{x} dx \leqslant$ $\left( \int_1^2 \sin^2 x dx \right)^{\frac 12}$ $\left( \int_1^2 \frac {dx}{x^2} \right)^{\frac 12}$i.e. $\int_1^2 \frac{\sin x}{x} dx \leqslant
\frac 12 \sqrt{1-\frac 12 \sin 4 + \frac 12 \sin 2}$



   

Example 15.0.3   Examples of inequalities which are consequences of the triangular inequality:


$V=\mathbb{R} ^2:$

$\forall x_1,x_2,y_1,y_2 \in \mathbb{R} $: $\sqrt{(x_1+y_1)^2+(x_2+y_2)^2} \leqslant \sqrt{x_1^2+x_2^2)}+\sqrt{(y_1^2+y_2^2)}$ y1=1,y2=10

$\Downarrow$

$\sqrt{(x_1+1)^2+(x_2+10)^2} \leqslant \sqrt{x_1^2+x_2^2)}+\sqrt{101}$






V=C[0,1]: $\left( \int_0^1 (f(x)+g(x))^2 dx \right)^{\frac 12}
\leqslant \left( \int_0^1 f(x)^2 dx \right)^{\frac 12}
+\left( \int_0^1 g(x)^2 dx \right)^{\frac 12}$ $\sqrt{\left( \int_0^1 (x + \sin x)^2 dx \right)} \leqslant
\underbrace{\sqrt{...
...\frac 12 \sin 2}}
+ \underbrace{\sqrt{\int_0^1 x^2 dx }}_{\frac {1}{\sqrt{3}}}$ $\left( \int_0^1 (x + \sin x)^2 dx \right)^{\frac 12}
\leqslant \frac{1}{\sqrt{2}}\sqrt{1- \frac 12 \sin 2}
+ \frac {1}{\sqrt{3}}$
    


   


$V=\mathbb{R} ^3:$ $\lambda_1=3, \lambda_2=2, \lambda_3=1$. $\sqrt{3(x_1+y_1)^2+2(x_2+y_2)^2+(x_3+y_3)^2}
\leqslant
\sqrt{3x_1^2+2x_2^2+x_3^2} +\sqrt{3y_1^2+2y_2^2+y_3^2}$ y1=y2=y3=1

$\Downarrow$

$\sqrt{3(x_1+1)^2+2(x_2+1)^2+(x_3+1)^2}
\leqslant
\sqrt{3x_1^2+2x_2^2+x_3^2} +\sqrt{6}$ x3=y1=y2=y3=1

$\Downarrow$

$\sqrt{3(x_1+1)^2+2(x_2+1)^2+4}
\leqslant \sqrt{3x_1^2+2x_2^2+x_3^2} +\sqrt{6}$






V=C[1,2]: $\left[ \int_1^2 \left( x+\frac{1}{x} \right)^2 dx \right]^{\frac 12}
\leqslant...
...2 dx \right)^{\frac 12}
+\left( \int_1^2 \frac {1}{x^2} dx \right)^{\frac 12}$i.e. $\left[ \int_1^2 \left( x+\frac{1}{x} \right)^2 dx \right]^{\frac 12}
\leqslant \sqrt{\frac 73} + \sqrt{\frac 12}$.




next up previous contents
Next: About this document ... Up: No Title Previous: The theorem of Pythagoras.
Noah Dana-Picard
2001-02-26