Functional analysis approach::Rigged Hilbert space


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  

Functional analysis approach
PREVIOUS: MotivationNEXT: Formal definition (Gelfand triple)