A linearizing operator or projection is a device which converts nonlinear information into linear form. A well-known example of a linearizing projection is the Shapley value, both in the descrete case (Shapley, 1953) and the continuous case (Aumann and Shapley, 1974). A linearizing projection usually satisfies certain axioms of rationality which insure that it is the "unique, fair" allocation or distribution. Thus it is an admiriable bookkeeping device because bookkeeping must be linear. In Ruckle (1982) a first attempt was made to treat the Aumann-Shapley theory of values in the setting of functionals defined in an arbitrary Banach space E. This paper continues the effort begun in Ruckle (1982) by constructing three global theories of linearizing projections (or x0-value). These theories are called "global" because they refer to spaces of functionals which are defined on the entire Banach space E. In Section 6 we shall describe what we mean by a "local" theory and explain why such theories are needed.