Rigged Hilbert space

In mathematics and physics, a rigged Hilbert space (Gelfand triple, nested Hilbert space, equipped Hilbert space) is a construction which can enlarge a Hilbert space to a bigger space containing additional objects which are not in the Hilbert space but which one would like to think of alongside the Hilbert space. For example, in the quantum mechanical description of a non-relativistic particle using the Hilbert space of square-integrable functions on the real line, eigenstates of the position and momentum operators are not in the Hilbert space, but are in a suitably defined rigged Hilbert space. Informally, the term "rigged" means that the Hilbert space has been equipped to do more than it otherwise could, in analogy with rigging a boat.

This construction is designed to link the distribution and square-integrable aspects of functional analysis. Such spaces were introduced to study spectral theory. They bring together the 'bound state' (eigenvector) and 'continuous spectrum', in one place.

Using this notion, a version of the spectral theorem for unbounded operators on Hilbert space can be formulated. "Rigged Hilbert spaces are well known as the structure which provides a proper mathematical meaning to the Dirac formulation of quantum mechanics."