ABSTRACT
The richness of complex analysis is an obvious motivation for the development of a formal theory of complexifications of real Banach spaces. The idea of complexifying a real Banach space has been considered in the past by many authors trying to solve specific problems in a real setting by using complex techniques. The investigation of analytic mappings on a real Banach space is one situation in which the complex techniques play a crucial role. Using a standard procedure, it is possible to extend multilinear mappings and polynomials in a unique way to the complexification of the space where they are defined. If in addition we can keep control on the norm of the extension, then results for complex Banach spaces can be interpreted in the real case. Complexification techniques have been used by several authors; see for instance A. Alexiewicz and W. Orlicz [1], C. Benitez, Y. Sarantopoulos and A. Tonge [2], J. Bochnak [3], J. Bochnak and J. Siciak [4], A. E. Taylor [22] and Verbitskii [24]. Using functional calculus, one can define a procedure to obtain the complex version of any real lattice (see [[5], p. 326]). A unified treatment of procedures for complexifying general real Banach spaces has been developed recently in two independent works by P. Kirwan [12] on the one hand, and by G. Munoz, Y. Sarantopoulos and A. Tonge [17] on the other. In this paper we present a survey of the most interesting results on complexifications.
