Uniform continuity of the product of real functions defined on a metric space

SEMINARIO DEL DOCTORADO DE ESTADÍSTICA, OPTIMIZACIÓN Y MATEMÁTICA APLICADA. CURSO 2017-2018

Título: Uniform continuity of the product of real functions defined on a metric space

Ponente: Gerald Beer (California State University Los Angeles)

Fecha: Viernes 3 de noviembre de 2017 a las 13:00 horas

Lugar: Sala de seminarios,  Instituto Universitario de Investigación CIO, Edificio Torretamarit, Universidad Miguel Hernández (Campus de Elche)

Resumen:

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.