The Machado--Bishop theorem in the uniform topology

TL;DR

This paper extends the Machado--Bishop theorem to bounded weighted vector-valued functions in the uniform topology, providing new formulas, applications, and insights into the structure of vector-valued function spaces.

Contribution

It introduces an analogue of Machado's distance formula for bounded functions and explores applications like Bishop--Stone--Weierstrass theorems and function space decompositions.

Findings

Derived a new distance formula for bounded vector-valued functions

Established results on the splitting of function spaces as tensor products

Discussed extensions of continuous vector-valued functions

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.