physics in the rigged Hilbert space is the complex eigenvector expansion. This is the generalization of the elementary basis vector expansion of a 3-dimensional vector, ${\bf x}=\sum_{i=1,2,3}{\bf e}_i({\bf e}_i\cdot{\bf x})=\sum{\bf e}_i\cdot x_i$ to the expansion of vectors $\phi^+\in\Phi_-$ using as basis vectors the generalized eigenvectors $|z_{R_i}^-\rangle$ and $|z^-\!\rangle^{(k)}$ of self-adjoint operators ${H}$ with complex eigenvalues $z_{R_i}$, and $z$, respectively.

Earlier developments towards this generalization were the fundamental theorem of linear algebra which states that for every self-adjoint operator $H$ in a $n$-dimensional Euclidian space ${\cal H}_n$ there exists an orthonormal basis $e_i\ldots e_n$ in ${\cal H}_n$ of eigenvectors $He_i=E_ie_i$. I.e., $f\in{\cal H}_n$ can be written $f=\sum_{i=1}^ne_i(e_i,f)$. This theorem generalizes to the infinite dimensional Hilbert space ${\cal H}$, but only for self-adjoint operators $H$ which are completely continuous (also called compact operators which include Hilbert-Schmidt, nuclear, traceclass operators). For an arbitrary self-adjoint operators $H$ one has to go outside the space to find a complete basis system of eigenvectors (generalized).

The [*first*]{} step in this direction is the Dirac basis vector expansion which in mathematical terms is called the nuclear spectral theorem. It states that for every $\phi\in\Phi$ $$\phi=\int^{+\infty}_0\,dE\;|E^+\rangle{\langle}^+E|\phi^+ \rangle\;+\;
  \sum_n\,|E_n)(E_n|\,\phi\, )\quad
\mbox{for }\;\phi \in{\Phi}$$ In here, $|E_n)$ are the discrete 