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

[Mampac]

Hi,

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

However, I feel like my proof is low-key incomplete. I first state that,
[math]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)[/math]Then
[math]||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|\}[/math]By triangle inequality, we can show that everything except for the first term in [imath]\max[/imath], is indeed upper-bounded, since [imath]|x + y| \leq |x| + |y|[/imath].
For the first term, if we expand, [imath]|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|[/imath].

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

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

 

[blamocur]

According to the definition, if [imath]||x||=0[/imath] then [imath]x=0[/imath] -- does it hold in your case?

 

[Mampac]

According to the definition, if [imath]||x||=0[/imath] then [imath]x=0[/imath] -- does it hold in your case?

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

 

[blamocur]

For [imath]n=3[/imath]: if [imath]x = (1,2,0)[/imath] what is [imath]||x||[/imath] ?

 

[Mampac]

For [imath]n=3[/imath]: if [imath]x = (1,2,0)[/imath] what is [imath]||x||[/imath] ?

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

 

[Dr.Peterson]

You said,

I verified that [imath]||x|| \geq 0, ||x|| = 0 \text{ iff } x = 0[/imath], ...

But it appears that you only verified [imath]||x|| = 0 \text{\color{red} if } x = 0[/imath].

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