 The integral (over the domain of the field operator) of the product of the test function \(f(x)\) and the field operator \(\phi(x)\) serves to “smooth out” the field operator over its domain; a more colloquial description is that the field is “smeared out” over space or spacetime. It is postulated within the axiomatic approach that a quantum field operator \(\phi[f\)] may be represented as an unbounded operator on a separable Hilbert space \(\Eta\), and that \(\{\phi[f]: f\in \Sigma \}\) (the set of smoothed field operators associated with \(\phi(x))\) has a dense domain \(\Omega\) in \(\Eta\). The smoothed field operators are often referred to as operator-valued distributions, and this means that for every \(\Phi,\Psi \in \Omega\) there is an element of the space of distributions \(\Sigma^x\), the topological dual of \(\Sigma\), that may be equated to the expression \(\langle \Phi {\mid} \phi[\ ]{\mid}\Psi\rangle\). If \(\Omega'\) denotes the set of functions obtained by applying all polynomials of elements of \(\{\phi[f]: f\in \Sigma \}\) onto the unique vacuum state, then the axioms mentioned above entail that \(\Omega'\) is dense in \(\Eta\) (asymptotic completeness) and that \(\Omega'\subset \Omega\) (domain axiom). The elements of \(\Omega\) correspond to possible states of the elements of \(\{\phi[f]: f\in \Sigma \}\). Though only one field has been considered thus far, the formalism is easily generalizable to a countable number of fields with an associated set of countably indexed field operators \(\phi_k (x)\) – cf. (Streater and Wightman 1964).