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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 $f=(f_1(\cdot),f_2(\cdot))$:

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

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.

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What do you mean by "multivalued"? – Qiaochu Yuan May 11 '11 at 2:37
@QiaochuYuan I mean that the functions map into $\mathbb R^n$ instead of simply $\mathbb R$. Please feel free to edit the title if you think it is confusing. – hulp10 May 11 '11 at 2:49
It is confusing, since "multivalued function" is reserved for another thing, see – Bruno Stonek May 11 '11 at 3:14
@BrunoStonek Thanks for the edit and sorry for the confusion. – hulp10 May 11 '11 at 3:16
up vote 1 down vote accepted

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.

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Thanks for the answer. I was doing something wrong when proving the triangle inequality. – hulp10 May 12 '11 at 2:13

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 $L^\infty_\mathbb{C}(\mathbb{R}^m)$

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It's too late right now to even think straight, but I suspect that if we take $\inf \{ C\ge 0 : \|f(x)\| \le C \mbox{ for almost every } x\}$ and any $n$, then the same thing works. I'll think it over tomorrow. – Bruno Stonek May 11 '11 at 4:26
Thanks for the answer. For my application I only need continuous and bounded functions, but your answer it is still helpful. – hulp10 May 12 '11 at 2:15

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