## Functional analysis approach::Rigged Hilbert space

### ::concepts

Hilbert::space ''H''::gelfand Spaces::space''' ''i''::which Theory::mathbb Subseteq::dense**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

- <math> \Phi \subseteq H </math>

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

- <math>\phi\mapsto\langle v,\phi\rangle</math>

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:

- <math>\Phi \subseteq H \subseteq \Phi^*. </math>

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 <math>\mathbb R^n</math>)

- <math>H=L^2(\mathbb R^n),\ \Phi = H^s(\mathbb R^n),\ \Phi^* = H^{-s}(\mathbb R^n)</math>,

where <math>s>0</math>.

**Rigged Hilbert space sections**

Intro Motivation Functional analysis approach Formal definition (Gelfand triple) References

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