Need help to understand norms I have some problems to understand norms.
At university, my task was to solve the following question:
$$
Is\;in\;\mathbb{R}^n (n\ge2)\;a\;norm\;defined\;by\;the\;following\;equations?\\
(a)\;\;f_1(x) = |x_1+x_2|\\
(b)\;\;f_2(x) = \sum_{i=0}^n(x_i^2)
$$
At first, i tried to figure out, how a norm is defined.
The charakteristics of a norm I found out are the following:
$$
Norm\;of\;x:= ||x||= \sqrt{x\cdot x}\\
positive\;definite:=||x|| = 0 \rightarrow x = 0\\
homogeneity:=||\lambda x|| = |\lambda|\;||x||\\
triangle\;inequality := ||x+y|| \le ||x||+||y||
$$
If all of them are true, a norm is defined as far as I found out.
The excercise with the sum was easier to solve since the homogeneity was not valid.
Subexcercise a made me some problems since the beginning of my argumentation was wrong. Our exercise instructors unfortunately do not keep on looking on the rest when we make a fault.
I tried to solve the subexcercise (a) as the following:
$$
positive\;definite:=||x|| = 0 \rightarrow x = 0\\
||x_1+x_2|| = 0\;means\; x_1=0=x_2 \rightarrow x=0\;true.
$$
This seemed to be the part where I made a mistake. But why is the statement above wrong? Unfortunately I don't see it.
 A: The definition $\|x\|=\sqrt{x\cdot x}$ is for a particular norm on $\mathbb{R}^n$. In your exercise you have two other functions defined on $\mathbb{R}^n$.
You have to verify whether the functions so defined have the required properties.
For $f_1(x)$ the positive definedness fails: consider the vector
$$
x=(1,-1,\underbrace{0,\dots,0}_{\text{$n-2$ zeros}})
$$
Then $f_1(x)=0$, but $x\ne0$.
What property fails for $f_2$? Hint: try homogeneity.

There's nothing special or magic in the symbol $\|\,{\cdot}\,\|$. It's just a symbol usually employed to denote a function that we already know it's a norm.
If $V$ is a vector space and $f\colon V\to \mathbb{R}$ is a function, then $f$ is a norm on $V$ provided:


*

*$f(v)\ge 0$, for all $v\in V$;

*$f(v)=0$ implies $v=0$;

*$f(\alpha v)=|\alpha|f(v)$ for all $v\in V$ and $\alpha\in\mathbb{R}$;

*$f(v+w)\le f(v)+f(w)$, for all $v,w\in V$.


One can define several norms on $\mathbb{R}^n$; for instance
$$
f(v)=\sum_{i=1}^n |v_i|,\quad\text{for $v=(v_1,v_2,\dots,v_n)$}
$$
Usually one writes $f(v)=\|v\|_1$. Another one is
$$
g(v)=\sqrt{v\cdot v},\quad\text{where $v\cdot v$ is the usual scalar product}
$$
and this is frequently written as $g(v)=\|v\|_2$ or just $\|v\|$.
Another example:
$$
h(v)=\max\{|v_1|,|v_2|,\dots,|v_n|\},\quad\text{for $v=(v_1,v_2,\dots,v_n)$}
$$
and this is frequently written $\|v\|_\infty$.
In your exercise, $f_2(v)=\sum_{i=1}^n v_i^2$ doesn't define a norm: if $v=(1,1,\dots,1)$ and $\lambda=2$, you have
$$
f_2(\lambda v)=f_2(2,\dots,2)=4n
$$
while
$$
|\lambda| f_2(v)=2f_2(1,1,\dots,1)=2n
$$
and they're not equal, since $n>0$.
