A Brief Exposition of the Space of Relatively Bounded Nonlinear Operators

Abstract

This note aims to present some rudimentary aspects of relatively bounded nonlinear (not necessarily linear) operators. To be precise, we prove that the set of all nonlinear operators between two normed linear spaces that are relatively bounded with respect to a fixed operator between these spaces itself is a normed linear space under a suitable norm. Furthermore, it is shown that if the codomain is a Banach space, then the space of all relatively bounded nonlinear operators is also complete. Some elementary results on the algebraic structure of the space of all nonlinear operators that are bounded with respect to a fixed operator are also recorded. The results reported in this note illustrate that with some predictable minor modifications, the fundamental classical results on the space of all bounded linear operators can be carried over to a more general setting. Here the word `general' is used in a double sense that the boundedness is generalized to relative boundedness and the restriction of linearity is dropped to include nonlinear maps.