Rigged Hilbert space

A Hilbert space containing a linear, everywhere-dense subset , 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 and a chain of continuous imbeddings (using the standard identification ). The most interesting case is that in which is a nuclear space. The following strengthening of the spectral theorem for self-adjoint operators acting on is true: Any self-adjoint operator mapping continuously (in the topology of ) onto itself possesses a complete system of generalized eigenfunctions ( is a set of indices), i.e. elements such that for any ,

where the set of values of the function , , is contained in the spectrum of (cf. Spectrum of an operator) and has full measure with respect to the spectral measure , , , of any element . The completeness of the system means that for any , , for at least one . Moreover, for any element , its expansion with respect to the system of generalized eigenfunctions 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

is a system of generalized eigenfunctions of the differentiation operator, acting on , arising under the natural rigging of this space by the Schwartz space (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)

Comments

A rigged Hilbert space is also called a Gel'fand triple. Occasionally one also finds the phrases nested Hilbert space, or equipped Hilbert space.