$X=\mathbb{R^2}$ and $f:\mathbb{R^2} \to {\mathbb{R}}$ defined by $f(x) =(\sum_{i=1}^{2}{|x_i|^{1/2}})^2$ , where $x=(x_1, x_2)\in {\mathbb{R^2}}$.
Does the function $f$ define a norm on $\mathbb{R^2}$ ?
My attempt:
I) $f$ satisfy Positivity. i.e. $f(x) \ge 0$ $\forall x \in \mathbb{R^2}$.
II) Definiteness is also true. i.e $f(x) =0$ iff $x=0$.
III) $f(\lambda x)=|\lambda|f(x)$ , $\forall x\in \mathbb{R^2} $ and $\forall \lambda \in \mathbb{R}$
IV) I find a counter example for triangle inequality.
$x=(1, 0) $ and $y=(0, 1) $ , then $f(x+y) =(\sqrt{|1+0|}+\sqrt{|0+1|})^2=4$
$f(x) =(\sqrt{1+0})^2=1$
$f(y) =(\sqrt{0+1})^2=1$
$f(x+y) =4>1+1=f(x)+f(y)$
Hence, triangle inequality fails to hold for this two particular points $x$ and $y$.
Is this proof correct? How to think quickly that the triangle inequality doesn't holds? How to produce counter example quickly? Thanks.