G. Beer
A fundamental extension theorem of McShane states that a bounded real-valued uniformly continuous function defined on a nonempty subset $A$ of a metric space $\langle X,d \rangle$ can be extended to a uniformly continuous function on the entire space. In the first half of this note, we obtain McShane's Theorem from the simpler fact that a real-valued Lipschitz function defined on a nonempty subset of the space has a Lipschitz constant preserving extension to the entire space. In the second half of the note, we use McShane's theorem to give an elementary proof of the equivalence of the most important characterizations of metric spaces in which the real-valued uniformly continuous functions form a ring. These characterizations of such a basic property, due to Cabello-Sánchez and separately Bouziad and Sukacheva, are remarkably recent.
Palabras clave: extension theorems, Lipschitz function, uniformly continuous function, pointwise product of uniformly continuous functions
Programado
GT11-5 MA-5 Optimización Continua. Homenaje a Marco Antonio López
6 de septiembre de 2019 12:40
I2L7. Edificio Georgina Blanes