Functionals (i.e., functions of functions) are widely used in quantum field theory and solid-state physics. In this paper, functionals are given a rigorous mathematical framework and their main properties are described. The choice of the proper space of test functions (smooth functions) and of the relevant concept of differential (Bastiani differential) are discussed. The relation between the multiple derivatives of a functional and the corresponding distributions is described in detail. It is proved that, in a neighborhood of every test function, the support of a smooth functional is uniformly compactly supported and the order of the corresponding distribution is uniformly bounded. Relying on a recent work by Dabrowski, several spaces of functionals are furnished with a complete and nuclear topology. In view of physical applications, it is shown that most formal manipulations can be given a rigorous meaning. A new concept of local functionals is proposed and two characterizations of them are given: the first one uses the additivity (or Hammerstein) property, the second one is a variant of Peetre’s theorem. Finally, the first step of a cohomological approach to quantum field theory is carried out by proving a global Poincaré lemma and defining multi-vector fields and graded functionals within our framework.
In this article, all manifolds shall be assumed to be paracompact.
Since an open set is absorbing (Ref. 30, p. 80), for every x ∈ U and every ∈ E there is an ϵ > 0 such that x + ∈ U if |t| < ϵ. Thus, f(x + ) is well defined for every t such that |t| < ϵ.
Since all manifolds in this article are assumed to be paracompact, for every property T on the set of all open subsets on M, provided that (i) open subsets of subsets satisfying T satisfy T and (ii) every point of M admits a neighborhood that satisfies property T, there exists an open cover made of subsets satisfying the property T that admits a partition of unity relative to it.
That is, .