Let $\mathcal{C}^k(\mathbb{R},\mathbb{R}^n)$ denote the space of $k$-times differentiable $\mathbb{R}^n$-valued functions on $\mathbb{R}$ equipped with the topology of uniform convergence of derivatives of order $1,\dots,k$ on every compact. Moreover let $\mathcal{B}(\mathcal{C}^k(\mathbb{R},\mathbb{R}^n))$ be the Borel $\sigma$-algebra generated by the open sets in $\mathcal{C}^k(\mathbb{R},\mathbb{R}^n)$ equipped with the same topology.
Consider the linear differential operator \begin{align} R : &\ \mathcal{C}^k(\mathbb{R},\mathbb{R}^n)\to \mathcal{C}^0(\mathbb{R},\mathbb{R}^n),\\ &f(t)\mapsto r_k\frac{\mathrm{d}^kf}{\mathrm{d}t^k}+r_{k-1}\frac{\mathrm{d}^{k-1}f}{\mathrm{d}t^{k-1}}+\cdots+r_0f, \quad r_i\in\mathbb{R}^{n\times n}, i=0,\dots,k \end{align}
- Is $R$ a continuous operator?
- Is $R$ a measurable operator from $(\mathbb{R}^n,\mathcal{B}(\mathcal{C}^k(\mathbb{R},\mathbb{R}^n)))$ to $(\mathbb{R}^n,\mathcal{B}(\mathcal{C}^0(\mathbb{R},\mathbb{R}^n)))$?
(If not, what about the case in which $R$ is defined on the space of smooth functions, i.e. $R\colon \mathcal{C}^\infty(\mathbb{R},\mathbb{R}^n)\to \mathcal{C}^\infty(\mathbb{R},\mathbb{R}^n)$?)