[Thm]: Hölder’s Inequality

Let $a_j,b_j\geq0$, $\forall j=1,...,n$, and $p,q>1$ s.t. ${1\over p}+{1\over q}=1$. Then

$$ \sum_{j=1}^n a_jb_j\leq \left(\sum_{j=1}^n a_j^p\right)^{1/p} \left(\sum_{j=1}^n b_j^q\right)^{1/q} $$

可以這麼記 $|\langle f,g \rangle|_1\leq\|f\|_p\cdot\|g\|_q$

• [pf]:

For $p\geq1$, $v=(v_1,...,v_n)\in\mathbb{R}^n$, define

$$ \|v\|p=\left(\sum{j=1}^n|v_j|^p\right)^{1/p} $$

For $x,y\in\mathbb{R}^n$, define

$$ <x,y>=\sum_{j=1}^n x_jy_j $$

[Lemma]: Dual Norm Property

For $x\in\mathbb{R}^n$, 且 $p$ is conjugate number of $q$ and vice versa. (i.e. $p,q>1$ s.t. ${1\over p}+{1\over q}=1$), 則

$$ \begin{align} \|x\|p=\sup{\|y\|_q=1}\{|\langle x,y\rangle |\} \end{align} $$

• [pf]:

[Thm]: Minkowski Inequality

Assume $p\geq1$. Then for any $x,y\in\mathbb{R}^n$ we have

$$ \|x+y\|_p\leq\|x\|_p+\|y\|_p $$

• [pf]:

[Def]: Normed Vector Space

Let $V$ be a vector space over $\mathbb{R}$. If mapping $N:V\rightarrow\mathbb{R}$ which satisfies the following conditions:

1. $N(v)\geq0$, $\forall v\in V$ and $N(v)=0 \Longleftrightarrow v=0$
2. $N(v+w)\leq N(v)+N(w),\forall v,w\in V$
3. $N(r\cdot v)=|r|N(v),\forall r\in\mathbb{R},v\in V$

Then the mapping is called a norm on $V$ and the pair $(V,N)$ is called a normed vector space

[例]:

On $\mathbb{R}^n$ we define $\|v\|_p=\left(\sum_j |v_j|^p\right)^{1/p}$, then

1. $\|v\|_p\geq0,\forall v\in \mathbb{R}^n$, and $\|v\|_p=0 \Longleftrightarrow |v_j|^p=0,\forall j \Longleftrightarrow v=0$
2. $\|v+w\|_p\leq\|v\|_p+\|w\|_p$ by Minkowski inequality
3. $\|r\cdot v\|_p=|r|\cdot\|v\|_p$ easy to prove

Hence $\|\cdot\|_p$ is a norm