Calculus on complex Banach spaces

Abstract

By using the techniques of modern functional analysis, a variety of new concepts have been developed and new results proved which extend considerably the new calculus on complex Banach spaces developed by Sharma and Rebelo. The distinguishing feature of the new calculus is that in this calculus the more general concept of additivity replaces that of linearity in the Frechet calculus. It is proved that the space of continuous additive maps between two complex Banach spaces is the direct sum of the spaces of linear and semilinear maps between the two spaces. The Hahn-Banach theorem and the open mapping theorem which in their standard versions are valid for continuous linear functionals and functions are shown to hold also for the additive case. The concepts of the adjoint of an additive map, of a new kind of orthogonal complement of a subset of a Banach space, and of a balanced additive map in which the norms of the linear and semilinear components are equal are developed. It is then proved that the orthogonal complement of the range of an additive map equals the null space of its adjoint and if the additive map is a functional on a complex Hilbert space and is balanced, then the orthogonal complement of the null space of the functional equals the range of the adjoint. A generalization of the inverse function theorem is proved by using our version of the open mapping theorem and then used to establish the Lagrange multiplier theorem in the new calculus. A number of related results are also proved. The applications of the new calculus to physics are briefly described.