The Machado--Bishop theorem in the uniform topology

Abstract

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.

arXiv comment27 pages; To appear in Journal of Approximation Theory

References in corpus (1)

• A note on Machado-Bishop theorem in weighted spaces with applications