Rigged Hilbert space

In mathematics, a rigged Hilbert space (Gelfand triple, nested Hilbert space, equipped Hilbert space) is a construction 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."

Motivation
A function such as $$ x \mapsto e^{ix}, $$ is an eigenfunction of the differential operator $$-i\frac{d}{dx}$$ on the real line $R$, but isn't square-integrable for the usual (Lebesgue) measure on $R$. To properly consider this function as an eigenfunction requires some way of stepping outside the strict confines of the Hilbert space theory. This was supplied by the apparatus of distributions, and a generalized eigenfunction theory was developed in the years after 1950.

Functional analysis approach
The concept of rigged Hilbert space places this idea in an abstract functional-analytic framework. Formally, a rigged Hilbert space consists of a Hilbert space $H$, together with a subspace $Φ$ which carries a finer topology, that is one for which the natural inclusion $$ \Phi \subseteq H $$ is continuous. 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 $Φ^{*}$. The 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 $$\phi\mapsto\langle v,\phi\rangle$$ 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: $$\Phi \subseteq H \subseteq \Phi^*. $$

The most significant examples are those for which $Φ$ is a nuclear space; this comment is an abstract expression of the idea that $Φ$ consists of test functions and $Φ*$ of the corresponding distributions. Also, a simple example is given by Sobolev spaces: Here (in the simplest case of Sobolev spaces on $$\mathbb R^n$$) $$H = L^2(\mathbb R^n),\ \Phi = H^s(\mathbb R^n),\ \Phi^* = H^{-s}(\mathbb R^n),$$ where $$s > 0$$.

Formal definition (Gelfand triple)
A rigged Hilbert space is a pair $(H, Φ)$ with $H$ a Hilbert space, $Φ$ a dense subspace, such that $Φ$ is given a topological vector space structure for which the inclusion map $i$ is continuous.

Identifying $H$ with its dual space $H^{*}$, the adjoint to $i$ is the map $$i^* : H = H^* \to \Phi^*.$$

The duality pairing between $Φ$ and $Φ^{*}$ is then compatible with the inner product on $H$, in the sense that: $$\langle u, v\rangle_{\Phi\times\Phi^*} = (u, v)_H$$ whenever $$u \in \Phi\subset H$$ and $$v \in H = H^* \subset \Phi^*$$. In the case of complex Hilbert spaces, we use a Hermitian inner product; it will be complex linear in $u$ (math convention) or $v$ (physics convention), and conjugate-linear (complex anti-linear) in the other variable.

The triple $$ (\Phi,\,\,H,\,\,\Phi^*)$$ is often named the "Gelfand triple" (after the mathematician Israel Gelfand).

Note that even though $Φ$ is isomorphic to $Φ^{*}$ (via Riesz representation) if it happens that $Φ$ is a Hilbert space in its own right, this isomorphism is not the same as the composition of the inclusion $i$ with its adjoint $i*$ $$i^* i: \Phi\subset H = H^* \to \Phi^*.$$