Rigged Hilbert space

A rigged Hilbert space, or Gelfand triple, is a chain

\Phi\hookrightarrow H\hookrightarrow\Phi^\times

in which $H$ is a Hilbert space, $\Phi$ is a Hausdorff nuclear locally convex space continuously and densely embedded in $H$ , and $\Phi^\times$ is its continuous anti-dual with the strong dual topology. Taking the inner product on $H$ linear in its first variable, the second embedding sends $h$ to the conjugate-linear functional $\phi\mapsto\langle h,\phi\rangle_H$ . Density makes this embedding injective.

Test vectors and generalized vectors

The space $\Phi$ carries a topology finer than the Hilbert norm topology, encoding regularity or decay. Elements of $\Phi^\times$ act as generalized vectors, including distributions that do not belong to $H$ . The Hilbert space lies between these two scales and identifies ordinary vectors with continuous anti-linear functionals on the test space.

Operators and spectral analysis

If a densely defined operator on $H$ preserves $\Phi$ and acts continuously on its locally convex topology, duality extends it to $\Phi^\times$ . Generalized eigenvectors may then live in $\Phi^\times$ even when the operator has no eigenvectors in $H$ . Nuclearity supplies the compactness and kernel-theorem structure used in generalized spectral expansions.

Example and convention

The canonical example is

\mathcal S(\mathbb R^n)\hookrightarrow L^2(\mathbb R^n)\hookrightarrow\mathcal S(\mathbb R^n)^\times,

with Schwartz functions as test vectors and the anti-linear version of tempered distributions as generalized vectors. Complex conjugation of functional values identifies this with the conventional linear distribution space $\mathcal S'(\mathbb R^n)$ conjugate-linearly.

Some authors use rigged Hilbert space for any continuous dense embedding $\Phi\hookrightarrow H$ , without requiring $\Phi$ to be nuclear, and reserve nuclear Gelfand triple for the definition above. The nuclear convention is used here. For the smooth vectors of a finite-dimensional Lie-group representation, this extra condition is equivalent to the representation being trace class.

References

I. M. Gel′fand and N. Ya. Vilenkin, Generalized Functions, Volume 4: Applications of Harmonic Analysis, Academic Press, 1964; AMS Chelsea reprint. AMS publisher record. Relevant: Chapter I on nuclear spaces, rigged Hilbert spaces, and spectral analysis.
Michael Reed and Barry Simon, Methods of Modern Mathematical Physics, Volume I: Functional Analysis, Academic Press, 1972. Elsevier DOI record. Relevant: Chapter V on locally convex spaces and distributions.