 The dual pair \((\Phi , \Phi^x)\) possesses the means to represent important operators for quantum mechanics that are problematic in a separable Hilbert space, particularly the unbounded operators that correspond to the observables position and momentum, and it does so in a particularly effective and unproblematic manner. As already noted, these operators have no eigenvalues or eigenvectors in a separable Hilbert space; moreover, they are only defined on a dense subset of the elements of the space and this leads to domain problems. These undesirable features also motivated von Neumann to seek an alternative to the separable Hilbert space framework for quantum mechanics, as noted above. In a rigged Hilbert space, the operators corresponding to position and momentum can have a complete set of eigenfunctionals (i.e., generalized eigenfunctions). The key result is known as the nuclear spectral theorem (and it is also known as the Gelfand-Maurin theorem). One version of the theorem says that if A is a symmetric linear operator defined on the space \(\Phi\) and it admits a self-adjoint extension to the Hilbert space H, then A possesses a complete system of eigenfunctionals belonging to the dual space \(\Phi^x\) (Gelfand and Shilov 1977, chapter 4). That is to say, provided that the stated condition is satisfied, A can be extended by duality to \(\Phi^x\), its extension \(A^x\) is continuous on \(\Phi^x\) (in the operator topology in \(\Phi^x)\), and \(A^x\) satisfies a completeness relation (meaning that it can be decomposed in terms of its eigenfunctionals and their associated eigenvalues). The duality formula for extending \(A\) to \(\Phi^x\) is \(\braket{\phi}{A^x\kappa} = \braket{A\phi}{\kappa}\), for all \(\phi \in \Phi\) and for all \(\kappa \in \Phi^x\). The completeness relation says that for all \(\phi ,\theta \in \Phi\):