Openness and weak openness of multiplication in the space of functions of bounded variation

Let BV[0,1] and CBV[0,1] be spaces of functions of bounded variation and continuous functions of bounded variation, respectively, with the norm ‖f‖BV=|f(0)|+V01(f). We shall show that the multiplication is a weakly open operation in BV[0,1] and CBV[0,1]. Moreover, the multiplication is an open operation in BV[0,1] with the standard supremum norm. Some other properties of multiplication in function spaces are studied.