 A rigged Hilbert space is a dual pair of spaces \((\Phi , \Phi^x)\) that can generated from a separable Hilbert space \(\Eta\) using a sequence of norms (or semi-norms); the sequence of norms is generated using a nuclear operator (a good approximate meaning is an operator of trace-class, meaning that the trace of the modulus of the operator is finite). In the mathematical theory of topological vector spaces, the space \(\Phi\) is characterized in technical terms as a nuclear Fréchet space. To say that \(\Phi\) is a Fréchet space means that it is a complete metric space, and to say that it is nuclear means that it is the projective limit of a sequence of Hilbert spaces in which the associated topologies get rapidly finer with increasing n (i.e., the convergence conditions are increasingly strict); the term nuclear is used because the Hilbert-space topologies are generated using a nuclear operator. In distribution theory, the space \(\Phi\) is characterized as a test-function space, where a test-function is thought of as a very well-behaved function (being continuous, n-times differentiable, having a bounded domain or at least dropping off exponentially beyond some finite range, etc). \(\Phi^x\) is a space of distributions, and it is the topological dual of \(\Phi\), meaning that it corresponds to the complete space of continuous linear functionals on \(\Phi\). It is also the inductive limit of a sequence of Hilbert spaces in which the topologies get rapidly coarser with increasing n. Because the elements of \(\Phi\) are so well-behaved, \(\Phi^x\) may contain elements that are not so well-behaved, some being singular or improper functions (such as Dirac’s \(\delta\) function). \(\Phi\) is the topological anti-dual of \(\Phi^x\), meaning that it is the complete set of continuous anti-linear functionals on \(\Phi^x\); it is anti-linear rather than linear because multiplication by a scalar is defined in terms of the scalar’s complex conjugate.