An extension of the Aumann-shapley value concept to functions on arbitrary banach spaces

LetX denote a linear space of real valued functions defined on a subset of a Banach space such thatX containsE′ the dual space ofE as a subspace. Given a distinguished vectorx 0 inE anx 0-value (onX) is defined to be a projectionP fromX ontoE′ which satisfies the following two hypotheses: (VA) (PF)(x0)=Fx0 for allF inX; (VB) IfT is a continuous isomorphism fromE intoE such thatTx 0=x 0 thenP(F⋄T) = (PF) ⋄ T for allF inX. The existence and uniqueness of a value is established for two choices ofX, one of which is the space of polynomials in functional onE. The existence and partial uniqueness of a value is established on a third choice forX.