Are there $C^\infty$ norms in $\mathbb{R^n}$ aside from the 2 norm(Euclidean norm)? $C^\infty$ in $\mathbb{R^n} - 0$, of course.
The question I'm working on is the following: "Prove that there is a $C^\infty$ function $f: \mathbb{R^n} - {0} \rightarrow \mathbb{R}$ which is positive and positively homogeneous of degree k. Prove also that are such functions which are not polynomials."
Def.: A function $f:U \rightarrow \mathbb{R}$ is said positively homogeneous of degree $k$, $k \in \mathbb{R},\ k \neq0$, when $\forall t > 0\ , \forall x \in \mathbb{R^n}$ we have $f(tx) = t^k f(x)$. We also require that $U$ satisfy: $\forall x \in U, tx \in U\ \forall t>0 $ in order for the definition to make sense. e.g. Linear functions on $\mathbb{R^n}$ are positively homogeneous of degree 1.
My progress so far: declare $f(x) = |x|^k$ (euclidean norm), whence $f(tx) = |tx|^k = |t|^k|x|^k = t^k f(x)$ since $t > 0$. It solves the first part. But if k is even, $f$ would be a polynomial in n indeterminates. That's why I'm looking for a $C^\infty$ norm in $\mathbb{R^n} - 0$ aside from the Euclidean.
So... Any lights?
Thanks in advance. Henrique.