Norm-attaining functionals on $C(Q,X)$

Functionals (vector measures) defined on the space $C(Q,X)$ of continuous abstract functions (where $Q$ is a compact Hausdorff space and $X$ is a Banach space) and attaining their norm on the unit sphere are considered. A characterization of such functionals is given in terms of the Radon–Nikodym derivative of the vector measure with respect to the variation of the measure and in terms of analogs of the derivative. Applications to the characterization of finite-codimensional subspaces with the best approximation property are given. Similar results are obtained for the space $B(Q,\Sigma,X)$ of uniform limits of simple functions.