The vector space of real functions, defined on the set of all mappings of a finite set P into another finite set L, splits into a sum of orthogonal subspaces, one for each subset of P. The orthogonal projections onto these subspaces merely involve averaging operations. Certain linear functional identities are equivalents of k-representability, i.e., of location in the span of those subspaces that belong to subsets of cardinality k. Potential applications refer to complex systems where these results could be used to analyze empirically how their properties depend on properties of their components as well as on the interactions between them. Roughly this amounts to estimating the internal structure of a ‘‘black box’’ from measured properties.