The space $\mathscr E'(K)$ of distributions with support in the comact set $K$ is a closed subspace of $\mathscr E'(\mathbb R^d)$ of all distributions, the latter space being the dual (with the topology of uniform convergence on all bounded sets) of $C^\infty(\mathbb R^d)$.

General locally convex theory then tells you that $\mathscr E'(K)$ is the dual of the quotient $\mathscr E(K)=C^\infty(\mathbb R^d)/I$ where $I$ turns out to the the space of smooth functions $f$ such that all derivatives $\partial^\alpha f$ vanish on $K$.

A priori, this is just some abstract space, but Whitney's extension theorem characterizes $\mathscr E(K)$ as the space of Whitney jets $(f^{(\alpha)})_{\alpha\in \mathbb N_0^d}$ which are families of continuous functions on $K$ whose formal Taylor series behave like the Taylor series of smooth functions on $\mathbb R^d$.

The space $\mathscr E'(K)$ of distributions with support in the comact set $K$ is a closed subspace of $\mathscr E'(\mathbb R^d)$ of all distributions with compact support, the latter space being the dual (with the topology of uniform convergence on all bounded sets) of $C^\infty(\mathbb R^d)$.

General locally convex theory then tells you that $\mathscr E'(K)$ is the dual of the quotient $\mathscr E(K)=C^\infty(\mathbb R^d)/I$ where $I$ turns out to the the space of smooth functions $f$ such that all derivatives $\partial^\alpha f$ vanish on $K$.

A priori, this is just some abstract space, but Whitney's extension theorem characterizes $\mathscr E(K)$ as the space of Whitney jets $(f^{(\alpha)})_{\alpha\in \mathbb N_0^d}$ which are families of continuous functions on $K$ whose formal Taylor series behave like the Taylor series of smooth functions on $\mathbb R^d$.