Let $C(\mathbb{R}^n;\mathbb{R}^d)$ be the space of continuous functions with the uniform-convergence on compacts topology. What are function spaces $X$ building on $C(\mathbb{R}^n;\mathbb{R}^d)$?

• $X$ properly contains $C(\mathbb{R}^n;\mathbb{R}^d)$,
• The subspace topology on $C(\mathbb{R}^n;\mathbb{R}^d)$ (with respect to $X$) agrees with the topology of uniform convergence on compact subsets,
• The elements of $X$ are functions from $\mathbb{R}^n$ to $\mathbb{R}^d$
• $C(\mathbb{R}^n;\mathbb{R}^d)$ is dense in $X$.