An Integral Duality Formula

We establish an integral formula for the duality between multilinear forms/homogeneous polynomials and tensor products for dual spaces with the approximation property and for which the injective tensor products of their preduals is separable and does not contain a copy of `1. We deduce some multilinear Bishop-Phelps-type results. Although spaces of multilinear forms and homogeneous polynomials are dual spaces, multilinear analogues of the linear Bishop-Phelps Theorem [6] do not hold in general [2, 14]. However, positive results exist. See [5] for a complete account and a detailed description of the problem. For an n-linear form defined on a product of Banach spaces X1 × · · · ×Xn, two (in general different) types of norm attainment can be considered: as a linear functional defined on the projective tensor product X1⊗̂π · · · ⊗̂πXn or as an n-linear form on X1× · · ·×Xn. Under certain conditions, the duality between multilinear forms and tensor products can be expressed by means of an integral formula which will establish the equivalence of the two types of norm attainment and thus yield some multilinear Bishop-Phelps-type results. We note that the same formula, for Hilbert spaces, has already been applied to the study of other geometrical properties of spaces of tensors [11]. 1. Notation and terminology (see [10] and [19] for details). Given n (real or complex) Banach spaces X1, . . . , Xn, we denote by X1 ⊗ · · · ⊗ Xn their tensor product and by π and ε the projective and injective norms respectively. If X1⊗̂π · · · ⊗̂πXn is the completion of X1 ⊗ · · · ⊗ Xn under the projective norm, then we have (X1⊗̂π · · · ⊗̂πXn)∗ = L(X1, . . . , Xn), the space of continuous nlinear forms on X1 × · · · × Xn endowed with the supremum norm. We denote by LI(X1, . . . , Xn) = (X1⊗̂ε · · · ⊗̂εXn)∗ and LN (X1, . . . , Xn) the spaces of integral and nuclear n-linear forms on X1 × · · · × Xn respectively. Let ⊗n,sX be the n-fold symmetric tensor product of X. If we endow it with the topology inherited from X⊗π · · · ⊗πX and denote its completion by (⊗̂n,s,πX) then we have (⊗̂n,s,πX)∗ = Ls(X), the space of continuous symmetric n-linear forms on X × · · · ×X endowed with the supremum norm. We denote by P (X) the space of all continuous n-homogeneous polynomials on X endowed with the natural supremum norm. The space ⊗n,s,πX can be renormed such that P(nX) becomes its dual [10]. Indeed, denote by x the tensor x⊗· · ·⊗x. Every element u of ⊗n,sX can be expressed as a finite sum ∑k j=1 λjx n j . Define the symmetric projective norm of u by ||u||πs = inf    k ∑ j=1 |λj | ||xj || : u = k ∑