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.