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 $\left\{C\_{\omega\_i}(\mathbb{R}^n,\mathbb{R}^d)\right\}$.




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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.



Ultrabornological representation for the space of uniformly continuous functions?
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 $\tilde f(x)=f(x)-f(0)$.




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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.


