Relations between $C(K)$ and $C(X; K)$ Let $X$ be locally compact Hausdorff and $K \subseteq X$ compact.
Denote by $C(K)$ the space of continuous functions $f : K \to \mathbb{R}$
and by $C(X; K)$ the space of continuous functions $f : X \to \mathbb{R}$ with support in $K$. Both are Banach spaces when equipped with the supremum norm
and $C(X; K)$ can be considered as a subspace of $C(K)$ by restriction. The dual $C(K)'$ can then be considered as a subspace of $C(X; K)'$ by $T \mapsto (f \mapsto T f|_K)$.
Are the duals identical? (I don't think so, since $C(X; K)$ is not dense in $C(K)$.) If not, does the quotient resp. a complement $A$ with $A \oplus C(K)' = C(X;K)'$ be described more explicitly?
 A: 
The dual $C(K)'$ can then be considered as a subspace of $C(X; K)'$ by $T \mapsto (f \mapsto T f|_K)$.

In general, that map is not injective, so $C(K)'$ is not a subspace of $C(X;K)'$ in that sense.
If $K$ is open in $X$, then the restriction $f \mapsto f\lvert_K$ is an isometric isomorphism $C(X;K) \to C(K)$, so it may be that the duals are identical (modulo canonical identifications).
In general, for the restriction $\rho_K \colon C(X;K) \to C(K)$, we have
$$\operatorname{im} \rho_K = \{ f\in C(K) : f\lvert_{\partial K} \equiv 0\} = \ker r_{\partial K}^K,$$
where $r_{\partial K}^K \colon C(K) \to C(\partial K)$ is the restriction. We thus have an exact sequence
$$0 \to C(X;K) \xrightarrow{\rho_K} C(K) \xrightarrow{r_{\partial K}^K} C(\partial K) \to 0\tag{1}$$
of Banach spaces and continuous linear maps, and correspondingly the exact dual sequence
$$0 \to C(\partial K)' \xrightarrow{(r_{\partial K}^K)'} C(K)' \xrightarrow{(\rho_K)'} C(X;K)' \to 0.\tag{2}$$
Thus we can describe $C(X;K)'$ as a quotient of $C(K)'$,
$$C(X;K)' \cong C(K)'/(r_{\partial K}^K)'(C(\partial K)').$$
