  A common approach to the theory of nonlocal Poisson brackets, seen from the
operatorial point of view, has been to keep implicit the sets on which these
brackets act. In this paper we aim to explicitly define appropriate functional
spaces underlying to the theory of 1 codimensional weakly nonlocal Poisson
brackets, motivating the definitions, and to prove the validity in this context
of some classical results in the field. We start by introducing the spaces for
the local case, which will serve as building tools for those in the nonlocal
one. The definition and the study of these nonlocal functionals are the core of
this work; in particular we work out a characterization of the variational
derivative of such objects. We then translate everything to the level of
manifolds, defining a global version of the functionals, and introduce the
notion nonlocal Poisson brackets in this context. We conclude by applying all
the machinery to prove a theorem due to Ferapontov. This last application is
the natural conclusion of our discussion and shows that the spaces we introduce
are suitable objects to work with when studying topics in this theory.
