We point out that a proper use of the Hoeffding-ANOVA decomposition for symmetric statistics of finite urn sequences, previously introduced by the author, yields a decomposition of the space of square-integrable functionals of a Dirichlet-Ferguson process, written L²(D), into orthogonal subspaces of multiple integrals of increasing order. This gives an isomorphism between L²(D) and an appropriate Fock space over a class of deterministic functions. By means of a well-known result due to Blackwell and MacQueen, we show that each element of the nth orthogonal space of multiple integrals can be represented as the L² limit of U-statistics with degenerate kernel of degree n. General formulae for the decomposition of a given functional are provided in terms of linear combinations of conditioned expectations whose coefficients are explicitly computed. We show that, in simple cases, multiple integrals have a natural representation in terms of Jacobi polynomials. Several connections are established, in particular with Bayesian decision problems, and with some classic formulae concerning the transition densities of multiallele diffusion models, due to Littler and Fackerell, and Griffiths. Our results may also be used to calculate the best approximation of elements of L²(D) by means of U-statistics of finite vectors of exchangeable observations.