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When the set $C^{k,r}(\mathbb{R}^n,\mathbb{R})$ is equipped with the usual semi-norm topology (https://en.wikipedia.org/wiki/Hölder_condition) it is known as the Hölder space. However, how do we define the semi-norm topology on $C^{k,r}(\mathbb{R}^n,\mathbb{R}^m)$ when $m>1$?
Are they simply: $$ \|f\|_K :=\sum_{I=1}^m \|f_i\|_{K,k,r}, $$ where for any $g\in C^{k,r}(\mathbb{R}^n,\mathbb{R})$ we define the semi-norm $\|g\|_{K,k,r}:=\sup_{x\in K} \max_{0\leq |\beta|\leq k} \|D^{\beta}g(x)\| + \max_{x_i\in K,x_1\neq x_2}\frac{\|D^kg(x_1)-D^kg(x_2)\|}{\|x_1-x_2\|^r}$

Does anyone ever consider instead: $$ \|f\|'_K := \sup_{x\in K}\|f(x)\| + \max_{x_i\in K,x_1\neq x_2}\frac{\|f(x_1)-f(x_2)\|}{\|x_1-x_2\|} + \sum_{I=1}^m \|f_i\|_{K,k,r}? $$

ABIM
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1 Answers1

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These two families of semi-norms are equivalent. It is clear that $\|f\|_K \leq \|f\|_K'$ so I consider only the other inequality. The case $k = 0$ is easier so I consider only the case $k \geq 1$.

For this it will suffice to see that $\sup_{x\in K} \|f(x)\|, \sup_{x_i \in K, x_1 \neq x_2} \frac{\|f(x_1) - f(x_2)\|}{\|x_1 - x_2 \|} \lesssim \|f\|_K$.

For the first of these, note that from the definition it is clear that $$\sup_{x \in K} \|f(x)\|_1 = \sup_{x \in K} \sum_{i = 1}^m |f_i(x)| \leq \sum_{i = 1}^m \sup_{x \in K} |f_i(x)| \leq \|f\|_K.$$

Since all norms on $\mathbb{R}^m$ are equivalent, this gives the desired result.

Using the same equivalence of norms trick, for the second inequality it suffices to consider $$\sup_K \frac{\|f(x_1) - f(x_2)\|_1}{\|x_1 - x_2\|}$$ and hence even to control $$\frac{ |f_i(x_1) - f_i(x_2)|}{\|x_1 - x_2\|}$$ for arbitrary $i$. For $k > 0$, an application of Taylor's theorem tells you that this is controlled by the $\sup$ norms of the first derivatives, yielding the desired equivalence.

Rhys Steele
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