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arXiv · 2407.19925

The Machado--Bishop theorem in the uniform topology

Deliang Chen
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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.

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Deliang Chen. 2024-07-29. The Machado--Bishop theorem in the uniform topology. https://doi.org/10.1016/j.jat.2024.106085

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