Rigged Hilbert space

Summary

A rigged Hilbert space, or Gelfand triple, is a chain $\Phi \subset H \subset \Phi'$. Here $H$ is a Hilbert space, $\Phi$ is a dense subspace with a finer topology, and $\Phi'$ is the dual of $\Phi$. It gives a rigorous home to objects such as the Dirac delta and plane waves, which are not vectors of $H$. With a nuclear space $\Phi$, it carries the spectral theorem in terms of generalized eigenvectors.