## 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

$\Phi \subseteq H$

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 $s>0$.

Rigged Hilbert space sections
Intro  Motivation  Functional analysis approach  Formal definition (Gelfand triple)  References

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