View source for Rigged Hilbert space

A [[Hilbert space|Hilbert space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r0823401.png" /> containing a linear, everywhere-dense subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r0823402.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r0823403.png" /> and a chain of continuous imbeddings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r0823404.png" /> (using the standard identification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r0823405.png" />). The most interesting case is that in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r0823406.png" /> is a [[Nuclear space|nuclear space]]. The following strengthening of the spectral theorem for self-adjoint operators acting on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r0823407.png" /> is true: Any [[Self-adjoint operator|self-adjoint operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r0823408.png" /> mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r0823409.png" /> continuously (in the topology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234010.png" />) onto itself possesses a complete system of generalized eigenfunctions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234011.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234012.png" /> is a set of indices), i.e. elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234013.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234014.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234015.png" /></td> </tr></table>

where the set of values of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234017.png" />, is contained in the spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234018.png" /> (cf. [[Spectrum of an operator|Spectrum of an operator]]) and has full measure with respect to the [[Spectral measure|spectral measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234021.png" />, of any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234022.png" />. The completeness of the system means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234023.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234025.png" />, for at least one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234026.png" />. Moreover, for any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234027.png" />, its expansion with respect to the system of generalized eigenfunctions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234028.png" /> 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|Fourier integral]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234029.png" /></td> </tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234030.png" /> is a system of generalized eigenfunctions of the differentiation operator, acting on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234031.png" />, arising under the natural rigging of this space by the Schwartz space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234032.png" /> (cf. [[Generalized functions, space of|Generalized functions, space of]]). The same assertions are also correct for unitary operators acting on a rigged Hilbert space.

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



====Comments====
A rigged Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234033.png" /> is also called a Gel'fand triple. Occasionally one also finds the phrases nested Hilbert space, or equipped Hilbert space.