Multiplication in the space of functions of bounded variation

Let BV[0, 1] denote the Banach space of all functions on the unit interval that have bounded variation. When endowed with the pointwise product, BV[0, 1] becomes a Banach algebra. We will demonstrate that multiplication in BV[0, 1] is an open map. Using an observation of F. Nazarov (who has kindly permitted us to include it), we show that multiplication is not uniformly open and we thereby solve in the negative the problem of whether open bilinear maps are automatically uniformly open. (C) 2018 Published by Elsevier Inc.