Consider the Banach space $X=C_0(\mathbb{R}^-; \mathbb{R^n})$ of vector valued continuous functions which vanish at infinity, with the supremum norm.
I am studying a paper which makes some considerations on the dual of this space, and I have some problems understanding why the statements are true. My main two questions are the following:
1. The authors claim that the normed dual space $X^*$ of $X$ is the space of matrix-measures $M(\mathbb{R}^+; \mathbb{R^{n\times n}})$.
My only problem here is the change of intervals; why does the domain change from $\mathbb{R}^-$ to $\mathbb{R}^+$? Is this relevant?}
The duality pairing, in this situation, is given by $$\langle \mu, \psi \rangle = \int_{\mathbb{R}^+} [d\mu(\theta)] \, \psi(-\theta).$$
2. Then the authors say that $X'=\mathbb{R}^n \times L^1(\mathbb{R}^+; \mathbb{R}^{n\times n})$ is a closed subspace of the dual, and that the duality is given by $$\langle (c,g), \psi\rangle = \psi(0)\cdot c + \int_0^\infty \psi(-\theta)\cdot g(\theta)\, d\theta.$$
My main question here is that I do not know how to see $X'$ as a subspace of $X^*$; it should be up to some identification, but I cannot understand which, nor can I see how the duality simplifies to the expression above.
Edit: The paper which I am studying can be found here: http://www.sciencedirect.com/science/article/pii/S0022039611004165; (Equations with infinite delay: Blending the abstract and the concrete, by O. Diekmann, M. Gyllenberg)
If the pairing is defined by $\langle \mu, \psi \rangle = \int_{\mathbb{R}^+} [d\mu(\theta)] \, \psi(-\theta)$ and the function $\psi$ is supported on the negative semi-axis, then $\mu$ should be supported on the positive semi-axis, due to the $-$ sign in $\psi(-\theta)$. I suppose the authors have a reason for having the minus sign there, which will become apparent when this duality gets used.
What I don't understand is why the measures take values in $\mathbb{R^{n\times n}}$. This should be $\mathbb{R}^n$ in order for $\langle \mu, \psi \rangle $, as defined above, to be a scalar.
For the second question: a pair $(c, g)$ induces the measure $\mu = c\delta_0 + g\,dx$ where $\delta_0$ is the Dirac delta at $0$. This measure is in the dual space, and the pairing described above leads to the given formula, $$\int_{\mathbb{R}^+} [d\mu(\theta)] \, \psi(-\theta) = \psi(0)\cdot c + \int_0^\infty \psi(-\theta)\cdot g(\theta)\, d\theta$$ Also, the norm on the dual space is the total variation norm, which, when specialized to $\mu$ of the above kind, is equal to $$\|(c,g)\| = \|c\| + \int_0^\infty |g| \tag1$$ Since $L^1$ is complete, the product $(\mathbb{R}^n, L^1)$ is complete under the norm (1), and thus is a closed subspace of whatever larger space.