The Machado--Bishop theorem in the uniform topology

The Machado–Bishop theorem for weighted vector-valued functions vanishing at infinity has been extensively studied. In this paper, we give an analogue of Machado’s distance formula for bounded weighted vector-valued functions. A number of applications are given; in particular, some types of the Bishop–Stone–Weierstrass theorem for bounded vector-valued continuous spaces in the uniform topology are discussed; the splitting of $C(I\times J,X\otimes Y)$ as the closure of $C(I,X)\otimes C(J,Y)$ in different senses and the extension of continuous vector-valued functions are studied.