A Fundamental Property of Bilinear Operators

Abstract

In this paper, we address a fundamental question about the nature of bilinear operators first posed by Walter Rudin. While it has been known that there is not an open mapping theorem for bilinear operators in general, we show that bilinear operators enjoy an open mapping theorem, when the range has dimension three or less, and we address the simple, but crucial property that differentiates bilinear operators from their linear counterparts. We also give an example which shows that polynomials, in general do not enjoy an open mapping theorem, even when the range has dimension three.