Norms for space $V$ of vector-valued mappings $f: \mathbb R^m \rightarrow \mathbb R^n$ I wonder whether there are natural norms for the space $V$ of vector-valued functions that map $\mathbb R^m$ into $\mathbb R^n$.
Formally, let's define $V$ as the set of $f$ such that $f: \mathbb R^m \rightarrow \mathbb R^n$. If the answers restrict $V$ to only continuous and/or bounded functions, that is fine for me.
I have tried to extend the usual norms for functions that map into $\mathbb R$, but I can never show the triangle inequality. $||f+g||\leq ||f||+||g||$.
I have tried (for the case with $n=2$ and denoting $f=(f_1(\cdot),f_2(\cdot))$:


*

*$||f||=\sup_{x \in \mathbb R^m} \left\{\max\left\{|f_1|,|f_2| \right\} \right\}$

*$||f||=\max\{\sup_{x\in\mathbb R^m}|f_1|,\sup_{x\in\mathbb R^m}|f_2|\}$


None of them seem to work.
I asked the question generally, but I am particularly interested in the case with $m=n=2$. Thanks for your suggestions.
Note: This question arises after the helpful comments in this other question.
 A: For any $m$, considering $\mathbb{R}^2\simeq \mathbb{C}$ then the space of measurable functions $\mathbb{R}^m \to \mathbb{C}$ which are essentially bounded, i.e. such that 
$\|f\|_\infty := \inf \{ C\ge 0 : |f(x)| \le C \mbox{ for almost every } x\} <\infty$
can be equipped with the $\infty$-seminorm defined above.
If you take the quotient by the kernel of $\|\cdot\|$, i.e. if you identify functions equal almost everywhere, then you get a norm on the space of essentially bounded measurable functions with any $m$ and $n=2$, denoted $L^\infty_\mathbb{C}(\mathbb{R}^m)$
A: The two norms you suggest are actually equal and do satisfy the triangle inequality. To see that they are equal, consider that taking a supremum of a finite set (i.e. $|f_1(x)|$ and $|f_2(x)|$ for a fixed $x$), is equal to taking the maximum, so one could rewrite both as
$\|f\| = \sup_{x \in \mathbb{R}^m, i = 1,2} |f_i(x)|.$
To see that it satisfies the triangle inequality, first check that the triangle inequality is satisfies for each fixed $x \in \mathbb{R}^m$. Then this also holds for the supremum over $x$, as inequalities are preserved under taking suprema.
Note that the space of bounded, continuous functions with this norm is a standard Banach space.
