It is known that continuous linear functionals on the space $C\_0({\mathbb{R}^n})$ are bounded Radon measures ${\cal M}({\mathbb{R}^n})$ where $C\_0({\mathbb{R}^n})$ is uniform closure of the space of continuous functions with compact support $C\_c({\mathbb{R}^n})$ (for reference e.g. Folland's book on real analysis chapter 7).


I am wondering if there is a similar more general result, namely: what would be a continuous linear function on $C\_0({\mathbb{R}^n})$ with values in some Banach space $E$? In analogy with the previous case where we had functionals, I assume it might be in the space of vector-valued bounded Radon measures ${\cal M}({\mathbb{R}^n};E)$?



On Radon measures with values in Banach space
The formulation is not quite clear---do you mean linear functionals on a space of Banach space valued functions or linear operators on scalar-valued functions with valued in a Banach space? In the former case, the representing measure takes its value in the dual and is regular with respect to the topology of uniform convergence on compacta. In the second case, the operator is represented by a measure with values in the bidual which is regulär with respect to the weak star topology. If you want a measure with values in the original you should suppose that the operator is weakly compact. A good reference is the classic on vector measures by Diestel and Uhl.


