Let $\Omega$ be open in $\mathbb{R}^n$, $1\leq k < \infty$, and $\{K_n\}_{n=1}^{\infty}$ a sequence of nested compact sets whose union is $\Omega$. Consider the nested Frechet spaces of $k$ times continuously differentiable functions with support in $K_n$, $\mathcal{C}_c^k(K_n)$, each equipped with the topology induced by the seminorms
$$\|f\|_{K}=\sup_{|p|\leq k}(\sup_{x\in K}|(\delta/\delta x)^p f(x)|)$$ where $p$ is a multi-index and $K$ are the compact sets in $K_n$.
Let $\mathcal{C}_c^k(\Omega)$ be the vector space of $k$ times continuously differentiable functions on $\Omega$ with compact support. I am trying to convince myself of the claim (e.g. Treves, Topological Vector Spaces, Distributions and Kernels, 1967, p132) that the LF topology induced on this space by $\mathcal{C}_c^k(K_n)$ is strictly finer than the topology induced by the space of all $k$ times continuously differentiable functions on $\Omega$, $\mathcal{C}^k(\Omega)$, with the topology defined by the family of seminorms
$$\|f\|_{K}=\sup_{|p|\leq k}(\sup_{x\in K}|(\delta/\delta x)^p f(x)|)$$
where $K$ are the compact sets in $\Omega$.
In that spirit, I am trying to think of, say, a linear functional that is continuous on each of $\mathcal{C}_c^k(K_n)$ but not on $\mathcal{C}^k(\Omega)$. This answer gives an example where an $LF$ topology is finer than an alternative topology on the space of sequences converging to zero. However, it would be nice to have an example in the $\mathcal{C}_c^k(\Omega)$ setting in order to justify use of the $LF$ topology when dealing with continuously differentiable functions with compact support. Any clues in this direction would be much appreciated.
P.S. Alternatively, it would be really great to have an example of a neighbourhood of zero in $\mathcal{C}_c^k(\Omega)$ with the $LF$ topology which is not to be found in the topology induced by $\mathcal{C}^k(\Omega)$.