# Gel’fand triple

## 1 Gel’fand (or Gelfand) triple

The basic idea is to equip a separable Hilbert space $\mathcal{H}$ with a dense Topological Subvector Space ($TVS$) of test functions in such a manner that the dual of the subspace of test functions $T_{f}$ enhances the Hilbert space $\mathcal{H}$ through embedding into a larger topological subvector space $T^{*}$; the elements of $T^{*}$ represent generalized eigenvectors for the continuous spectrum of normal-and possibly unbounded-linear operators $L$.

One begins the Gelfand construction with a Banach space $B$, or the more general TVS considered above, and denote the dual TVS of $B$ as $B^{*}$ so that $j:B\rightarrow\mathcal{H}$ is an injective bounded operator with dense image. One then considers a canonical isomorphism $i_{c}:\mathcal{H}\cong\mathcal{H}^{*}$ determined by the inner product given by the Riesz theorem. The Banach transpose, or dual of $j$ is the operator $j^{*}:\mathcal{H}^{*}\to B^{*}$.

Then, one has a composition morphism $k:\mathcal{H}\rightarrow B^{*}$ as $\mathcal{H}\to\mathcal{H}^{*}\to B^{*}$ , so that $k=j^{*}\circ i_{c}$ that can be employed to define the Gelfand triple as follows.

###### Definition 1.1.

A Gelfand triple is defined by the operator sequence:

$B\rightarrow\mathcal{H}\rightarrow B^{*},$

where $k$ is the composition morphism $k:\mathcal{H}\to B^{*}$ defined by $j^{*}\circ i_{c}$

### 1.1 Example:

An interesting example of a Gel’fand triple which is preserved by the Fourier transform (FT) is obtained when $B=\mathcal{S}(\mathcal{R}^{n})$ –the Schwartz space, $B^{*}=\mathcal{S}^{t}$-the space of tempered Schwartz distributions and $\mathcal{H}=L^{2}(\mathcal{R}^{n})$.

### 1.2 Acknowledgement

The formalism utilized here followed the http://ncatlab.org/nlab/show/Gelfand+triplepresentation by Urs Schreiber of the Gel’fand triple at http://ncatlab.org/nlab/show/HomePage$nLab$- a ‘steered’ wiki for Mathematics, Physical Mathematics and Philosophy.

### 1.3 References

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Title Gel’fand triple GelfandTriple 2013-03-22 19:22:51 2013-03-22 19:22:51 bci1 (20947) bci1 (20947) 9 bci1 (20947) Topic msc 81P10 rigged Hilbert space $\mathcal{S}$ ${\mathcal{S}}^{t}$ injective bounded operator $$ operator TVS topological subvector space canonical isomorphism