|3 noviembre, 2017|
Title: Uniform continuity of the product of real functions defined on a metric space
Speaker: Gerald Beer (California State University Los Angeles)
Date: 03/11/2017, 13:00h
Location: Sala de Seminarios (Edificio Torretamarit)
As is well-known, the pointwise product of two bounded real-valued uniformly continuous functions f and g defined on a metric space is uniformly continuous, but this condition is hardly necessary: let f(x) = g(x) = \sqrt (x) for x ≥ 0. In joint work with Som Naimpally (deceased) that appeared in Real Analysis Exchange in 2012, we produce necessary and sufficient conditions on a pair of uniformly continuous real-valued functions (f,g) defined on an arbitrary metric space so that their pointwise product is uniformly continuous. Our conditions are sufficient for any pair of real-valued functions, and are necessary for a class of pairs properly containing the uniformly continuous pairs. We precisely identify this class and describe it in various ways.
A different but equally interesting question is this: what are necessary and sufficient conditions on the internal structure of a metric space so that the product of each pair of uniformly continuous real-valued functions remains uniformly continuous? In joint work with Maribel Garrido and Ana Meroño appearing in Set-Valued and Variational Analysis and dedicated to Jon Borwein that in part revisits a recent result of Javier Cabello Sánchez appearing in Filomat, we show that this occurs exactly when the familiar bornology of Bourbaki bounded subsets coincides with a new larger bornology.