Showing that $||(x, y)||_1$ := $|x| + |y|$ is a norm

Question

Show that $||(x, y)||_1$ := $|x| + |y|$ is a norm on $\mathbb R^2$ := {($x, y$) : $x$ ∈ $\mathbb R$, $y$ ∈ $\mathbb R$}

I've came across this problem but I don't really know what to do. Like, if $||(x, y)||_1$ := $|x| + |y|$, doesn't that directly mean that it's a norm since it is always defined for any values of $x$ and $y$? Please I need help. Thank you

Answer 1

You have to show:

1. $||(x,y)||_1 \ge 0.$

2. $||(x,y)||_1 = 0 \iff (x,y)=(0,0).$

3. For $ \alpha \in \mathbb R$ we have $|| \alpha(x,y)||_1 = |\alpha | \cdot ||(x,y)||_1.$

4. $||(x,y)+(u,v)||_1 \le ||(x,y)||_1+||(u,v)||_1.$