# Ultrabornological representation for the space of uniformly continuous functions?

Let $$\{\omega_i\}_{i\in I}$$ be a non-empty set of increasing (not necessarily strictly) continuous functions preserving $$0$$. Then, for each $$i \in I$$ define the space $$C_{\omega_i}(\mathbb{R}^n,\mathbb{R}^d):= \left\{ f \in (\mathbb{R}^n,\mathbb{R}^d):\, \|f\|_{\omega_i,\infty}<\infty \right\} \mbox{ where } \|f\|_{\omega_i,\infty}:= \sup_{x \in \mathbb{R}^n} (\omega_i(\|x\|)+1)^{-1}\|f(x)\|.$$ Then this is a Banach space since the map $$f \mapsto f (\omega_i(\|x\|)+1)$$ is clearly an isometry with $$C_0(\mathbb{R}^n,\mathbb{R}^d)$$ for the sup-norm, and the maps between $$C_{\omega_i}(\mathbb{R}^n,\mathbb{R}^d)$$ and $$C_{\omega_j}(\mathbb{R}^n,\mathbb{R}^d)$$ can be defined similarly by rescaling analogously. This makes $$I$$ into a poset with $$i\leq j \mbox{ if and only if } \sup_{x \in \mathbb{R}^n}\omega_i(\|x\|) \leq \sup_{x \in \mathbb{R}^n}\omega_j(\|x\|).$$ Thus, we can define the LCS colimit of this co-cone $$\left\{C_{\omega_i}(\mathbb{R}^n,\mathbb{R}^d)\right\}$$.

Now for the question, if $$\{\omega_i\}_i$$ is taken to be the collection of all monotonically increasing and continuous functions identifying $$0$$ (i.e.: $$\omega(0)=0$$) then does $$\operatorname{co-lim}_i C_{\omega_i}(\mathbb{R}^n,\mathbb{R}^d)$$ contain all uniformly continuous functions?

Note: Here the colimit is in the LCS sense.

## 1 Answer

Maybe, I miss something, but the answer seems to be easy: If $$f:\mathbb R^n\to\mathbb R^d$$ is continuous with $$f(0)=0$$ you can define the weight function $$\omega(r)=\sup\{\|f(x)\|: \|x\|\le r\}$$ which is obviously increasing with $$\omega(0)=0$$. Moreover, it is continuous at $$r\ge 0$$ because of the uniform continuity of $$f$$ on the compact set $$\{x\in\mathbb R^n:\|x\|\le r+1\}$$. Then $$f\in C_\omega(\mathbb R^n, \mathbb R^d)$$ holds. The extra assumption $$f(0)=0$$ can be removed by applying this to $$\tilde f(x)=f(x)-f(0)$$.

This shows that $$C(\mathbb R^n, \mathbb R^d)$$ is the co-limit of $$\{C_\omega(\mathbb R^n, \mathbb R^d)\}$$ in the category of vector spaces. Whether this holds in LCS is a different question.

• You mean in the category of topological vector spaces? In that case $C(\mathbb{R}^n,\mathbb{R}^d)$ has compact-convergence topology or some (non-metrizable) final topology? Also, are these examples where, $I$ is say countable and fully ordered (besides the space of functions of moderate growth?)
– ABIM
Jul 31, 2020 at 8:57
• You wrote: Does the co-limit contain all uniformly continuous functions? -- This sounds very much like an algebraic question (where the co-limit is the union). Jul 31, 2020 at 10:31
• Ah, I see the issue; my error. But I'm mostly looking in the LCS category (I made an edit to help clarify), I'm sorry abou the confusion
– ABIM
Jul 31, 2020 at 10:46
• Note that a uniformly continuous function on a normed space has a linear growth, $\|f(x)\|\le a\|x\|+b$. So you don't need such a wide class of weights $\{\omega_i\}$. 0n the other hand, adding $1$ in the denominator in the definition of the norms means you do not care to much about being uniformly continuous or just continuous. Jul 31, 2020 at 14:38