I am reading this paper and I have a question regarding the following statement (p.8):
Let $\Omega \subset \mathbb{R}^d$ compact, then $C^1(\Omega, \mathbb{R}^k)^*,C(\Omega, \mathbb{R}^{k\times d})^*$ are the topological dual spaces of $C^1(\Omega, \mathbb{R}^k)$ and $ C(\Omega, \mathbb{R}^{k\times d})$.
Further $M(\Omega, \mathbb{R}^k)$ is the set of $k$-valued signed Borel-measures on $\Omega$. By Riesz-Markov-Theorem we know that $M(\Omega, \mathbb{R}^{k\times d}) = C(\Omega, \mathbb{R}^{k\times d})^*$.
Then the characterization $C^1(\Omega, \mathbb{R}^{k})^*$ is done as follows: For any $g \in M(\Omega, \mathbb{R}^k)$, we define the linear operator on $C^1(\Omega, \mathbb{R}^{k})$ as
$$\phi \mapsto \int_{\Omega} \langle \phi(x), g(dx)\rangle.$$
The author concludes that $M(\Omega, \mathbb{R}^{k,d}) \subseteq C^1(\Omega, \mathbb{R}^{k})^*$.
But in my opinion, it only shows that $M(\Omega, \mathbb{R}^{k}) \subseteq C^1(\Omega, \mathbb{R}^{k})^*$, since $g \in M(\Omega, \mathbb{R}^k)$. Maybe it works, if we take the operator $$ \phi \mapsto \int_{\Omega} \sum_{i=1}^d \langle \phi(x), g_i(dx)\rangle. $$ Why does that work? Thanks!