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 in the broad sense.^[vague] They bring together the 'bound state' (eigenvector) and 'continuous spectrum', in one place.

Motivation

A function such as

$${\displaystyle x\mapsto e^{ix},}$$

$${\displaystyle -i{\frac {d}{dx))}$$

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

$${\displaystyle \Phi \subseteq H}$$

$${\displaystyle \phi \mapsto \langle v,\phi \rangle }$$

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:

$${\displaystyle \Phi \subseteq H\subseteq \Phi ^{*}.}$$

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 ${\displaystyle \mathbb {R} ^{n))$)

$${\displaystyle H=L^{2}(\mathbb {R} ^{n}),\ \Phi =H^{s}(\mathbb {R} ^{n}),\ \Phi ^{*}=H^{-s}(\mathbb {R} ^{n}),}$$

${\displaystyle s>0}$

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

$${\displaystyle i^{*}:H=H^{*}\to \Phi ^{*}.}$$

The duality pairing between Φ and Φ^* is then compatible with the inner product on H, in the sense that:

$${\displaystyle \langle u,v\rangle _{\Phi \times \Phi ^{*))=(u,v)_{H))$$

${\displaystyle u\in \Phi \subset H}$

${\displaystyle v\in H=H^{*}\subset \Phi ^{*))$

The triple ${\displaystyle (\Phi ,\,\,H,\,\,\Phi ^{*})}$ 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*

$${\displaystyle i^{*}i:\Phi \subset H=H^{*}\to \Phi ^{*}.}$$