I have to prove the following assertion :
Let $V$ be a finit dimentional vector space with dimension $n$ over the field $K$ which is the field of real numbers or complex numbers. Let the map defined by:
$\mid \mid . \mid \mid _{\infty} \quad : v \mapsto \mid \mid v \mid \mid_{\infty}=\max\{\mid v_1\mid,…,\mid v_n \mid \} \quad \forall v \in V$ where $v_k \quad k=1,…,n $ are the componant of the vector $v$. Show that this map is a norm.
- I proved the posivity of this norm.
- I also proved that : $\mid \mid \lambda v \mid \mid _{\infty}=\mid \lambda \mid \mid \mid v \mid \mid \quad \forall v \in V, \forall \lambda \in K$
Now, I have to prove the triangle inequality. I have already done something but i am not sure what i have done is correct :
I have to prove that : $$\forall v, w \in V \quad \mid \mid v+w \mid \mid _{\infty}\leq \mid \mid v \mid \mid_{\infty} + \mid \mid w \mid \mid _{\infty}$$
I know that in the real or complex numbers, we have this inequality for all $x$ and $y$ : $\mid x+y\mid \leq \mid x \mid +\mid y\mid $
Therefore :
$v=(v_1,…,v_n)$, $w=(w_1,…,w_n)$ :
$$\forall k \in {1,…,n} \quad \mid v_k + w_k \mid \leq \mid v_k \mid +\mid w_k \mid $$
Then,
$$\max\{\mid v_1 +w_1\mid,… \mid v_n +w_n\mid \} \leq \max\{\mid v_1 \mid +\mid w_1 \mid ,…,\mid v_n\mid+\mid w_n \mid \}$$
Now the problem is : can i say that $$\max\{\mid v_1 \mid +\mid w_1 \mid ,…,\mid v_n\mid+\mid w_n \mid \} \leq \max\{\mid v_1\mid ,…\mid v_n\mid \}+\max\{\mid w_1\mid ,…\mid w_n\mid \}$$ which would prove the inequality ? Is it trivial ? If it is not, i don't know how to prove the last inequality…
Best Answer
It is very easy. All you need is $|v_i|+|w_i| \leq \max \{|v_1|,...,|v_n|\}+\max \{|w_1|,...,|w_n|\}$ for each $i$. This is true, right?