[Def]: Norm

On a vector space $X$ over $\mathbb{R}$, if $N$ is a function $N:X\rightarrow \mathbb{R}$, which satisfies the following conditions:

1. $N(x)\ge0$, $\forall x \in X$, $N(x)=0 \iff x=0$
2. $N(r\cdot x)=|r|\cdot N(x)$, $\forall r \in \mathbb{R}$, $x\in X$
3. $N(x+y)\le N(x)+N(y)$

Then $N$ is called a norm on $X$. Usually denotes $N(x) = \|x\|_N$

[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