Rigged Hilbert space

In mathematics, a rigged Hilbert space is a construction designed to link the distribution (test function) and square-integrable aspects of functional analysis. Such spaces were introduced to study spectral theory in the broad sense. They can bring together the 'bound state' (eigenvector) and 'continuous spectrum', in one place.

Formally, a rigged Hilbert space consists of a Hilbert space H, together with a subspace Φ which carries a finer topology. It is no loss to assume that Φ is dense in H for the Hilbert norm. We consider the inclusion of dual spaces H^* in Φ^*. Thr latter, dual to Φ in its 'test function' topology , is realised as a space of distributions or generalised functions of some sort, and the linear functionals on the subspace Φ of type φ -> for v in H are faithfully represented as distributions (because we assume Φ dense).

Now by applying the Riesz representation theorem we can identify H^* with H. Therefore the definition of rigged Hilbert space is in terms of a sandwich: H lies between Φ, a test function space, and Φ^*, its dual space of generalised functions.