Check validity of a vector norm; verifying ||x + y|| < ||x|| + ||y||

[Mampac]

Hi,

Given a vector $x = (x_1, x_2, ..., x_n)$ the vector norm
$$\max\{|2x_1 - x_2|, |x_3|, ..., |x_n|\}$$I verified that $||x|| \geq 0, ||x|| = 0 \text{ iff } x = 0$, also that $||\alpha x|| = |\alpha|||x||$, now I need to prove the triangle inequality, i.e. $||x + y|| < ||x|| + ||y||$, as the last property of a vector norm.

However, I feel like my proof is low-key incomplete. I first state that,
$$x = (x_1, x_2, ..., x_n) \\ y = (y_1, y_2, ..., y_n) \\ x + y = (x_1 + y_1, x_2 + y_2, ..., x_n + y_n)$$Then
$$||x|| = \max\{|2x_1 - x_2|, |x_3|, ..., |x_n|\} \\ ||y|| = \max\{|2y_1 - y_2|, |y_3|, ..., |y_n|\} \\ ||x + y|| = \max\{|2(x_1 + y_1) - (x_2 + y_2)|, |x_3 + y_3|, ..., |x_n + y_n|\}$$By triangle inequality, we can show that everything except for the first term in $\max$, is indeed upper-bounded, since $|x + y| \leq |x| + |y|$.
For the first term, if we expand, $|2(x_1 + y_1) - (x_2 + y_2)| = |2x_1 + 2y_1 - x_2 - y_2| = |2x_1 - x_2 + 2y_1 - y_2| \leq |2x_1 - x_2| + |2y_1 - y_2|$.

The fact that I compare only respective terms makes me feel off. Is that all?

I'm just unsure how to work with the $\max$ functions and how to prove upper bounds on them:/
Thanks for help

 

[blamocur]

According to the definition, if $||x||=0$ then $x=0$ -- does it hold in your case?

 

[Mampac]

According to the definition, if $||x||=0$ then $x=0$ -- does it hold in your case?

yes, since each term in the max function has an $x_i$, they all evaluate to $0$, and max itself evaluates to $0$, am I not right?
$$\max\{|2 \cdot 0 - 0|, |0|, ..., |0|\}$$

 

[blamocur]

For $n=3$: if $x = (1,2,0)$ what is $||x||$ ?

 

[Mampac]

For $n=3$: if $x = (1,2,0)$ what is $||x||$ ?

ugh, goddamit. my fault. you're totally right, thanks

 

[Dr.Peterson]

You said,

I verified that $||x|| \geq 0, ||x|| = 0 \text{ iff } x = 0$, ...

But it appears that you only verified $||x|| = 0 \text{\color{red} if } x = 0$.

To check the only if part, you need to suppose that $||x|| = 0$ and try to show that $x = 0$. That is, suppose that $\max\{|2x_1 - x_2|, |x_3|, ..., |x_n|\}=0$, which implies that $|2x_1 - x_2|, |x_3|, ..., |x_n|$ are all zero. But $|2x_1 - x_2|=0$ does not imply that $x_1= x_2=0$.