# Rigged Hilbert space

A Hilbert space $\mathcal{H}$ containing a linear, everywhere-dense subset $\Phi \subseteq \mathcal{H}$, on which the structure of a topological vector space is defined, such that the imbedding is continuous. This imbedding generates a continuous imbedding of the dual space $\mathcal{H}' \subseteq \Phi'$ and a chain of continuous imbeddings $\Phi \subseteq \mathcal{H} \subseteq \Phi'$ (using the standard identification $\mathcal{H}' = \mathcal{H}$). The most interesting case is that in which $\Phi$ is a nuclear space. The following strengthening of the spectral theorem for self-adjoint operators acting on $\mathcal{H}$ is true: Any self-adjoint operator $A$ mapping $\Phi$ continuously (in the topology of $\Phi$) onto itself possesses a complete system of generalized eigenfunctions $(F_{\alpha} \mid \alpha \in \mathfrak{A})$ ($\mathfrak{A}$ is a set of indices), i.e. elements $F_{\alpha} \in \Phi'$ such that for any $\phi \in \Phi$, $${F_{\alpha}}(A \phi) = \lambda_{\alpha} {F_{\alpha}}(\phi), \qquad \alpha \in \mathfrak{A},$$ where the set of values of the function $\alpha \mapsto \lambda_{\alpha}$, $\alpha \in \mathfrak{A}$, is contained in the spectrum of $A$ (cf. Spectrum of an operator) and has full measure with respect to the spectral measure ${\sigma_{f}}(\lambda)$, $f \in \mathcal{H}$, $\lambda \in \Bbb{R}$, of any element $f \in \mathcal{H}$. The completeness of the system means that ${F_{\alpha}}(\phi) \neq 0$ for any $\phi \in \Phi$, $\phi \neq 0$, for at least one $\alpha \in \mathfrak{A}$. Moreover, for any element $\phi \in \Phi$, its expansion with respect to the system of generalized eigenfunctions $(F_{\alpha} \mid \alpha \in \mathfrak{A})$ exists and generalizes the known expansion with respect to the basis of eigenvectors for an operator with a discrete spectrum.

Example: The expansion into a Fourier integral $$f(x) = \int_{\Bbb{R}} e^{i s x} \tilde{f}(s) ~ \mathrm{d}{s}, \qquad x \in \Bbb{R}, \qquad f,\tilde{f} \in {L^{2}}(\Bbb{R}),$$ $(x \mapsto e^{i s x} \mid s \in \Bbb{R})$ is a system of generalized eigenfunctions of the differentiation operator, acting on ${L^{2}}(\Bbb{R})$, arising under the natural rigging of this space by the Schwartz space $\mathcal{S}(\Bbb{R})$ (cf. Generalized functions, space of). The same assertions are also correct for unitary operators acting on a rigged Hilbert space.

#### References

 [1] I.M. Gel'fand, G.E. Shilov, “Some problems in the theory of differential equations”, Moscow (1958) (In Russian) [2] I.M. Gel'fand, N.Ya. Vilenkin, “Generalized functions. Applications of harmonic analysis”, 4, Acad. Press (1964) (Translated from Russian) [3] Yu.M. [Yu.M. Berezanskii] Berezanskiy, “Expansion in eigenfunctions of selfadjoint operators”, Amer. Math. Soc. (1968) (Translated from Russian)

A rigged Hilbert space $\Phi \subseteq \mathcal{H} \subseteq \Phi'$ is also called a Gel'fand triple. Occasionally one also finds the phrases nested Hilbert space, or equipped Hilbert space.