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\section*{Gelfand triple}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{functional_analysis}{}\paragraph*{{Functional analysis}}\label{functional_analysis}

[[!include functional analysis - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{basic_examples}{Basic examples}\dotfill \pageref*{basic_examples} \linebreak
\noindent\hyperlink{a_long_example}{A long example}\dotfill \pageref*{a_long_example} \linebreak
\noindent\hyperlink{isomorphisms}{Isomorphisms}\dotfill \pageref*{isomorphisms} \linebreak
\noindent\hyperlink{terminology_and_references}{Terminology and references}\dotfill \pageref*{terminology_and_references} \linebreak
\noindent\hyperlink{terminological_variants_and_history}{Terminological variants and history}\dotfill \pageref*{terminological_variants_and_history} \linebreak
\noindent\hyperlink{early_references}{Early references}\dotfill \pageref*{early_references} \linebreak
\noindent\hyperlink{modern_references_and_links}{Modern references and links}\dotfill \pageref*{modern_references_and_links} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \emph{Gel'fand triple} is a structure that equips a [[Hilbert space]] with a dense [[topological vector space|topological vector subspace]] of good ``test'' functions, so that the dual of the subspace of test functions enhaces the Hilbert space by embedding it into a larger [[TVS]] whose elements can be considered as generalized [[eigenvector]]s for the continuous [[spectrum of an operator|spectrum]] of [[normal operator|normal]] (possibly [[unbounded operator|unbounded]]) [[linear operator]]s.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

A \textbf{Gel'fand triple} is datum of the form

\begin{displaymath}
B\stackrel{J}\hookrightarrow H \stackrel{K}\hookrightarrow B^*
\end{displaymath}
where $H$ is a [[separable Hilbert space]], $B$ is a [[Banach space]] (or more general [[topological vector space]] (TVS)), $B^*$ is a dual TVS of $B$, $J:B\hookrightarrow H$ is an injective [[bounded operator]] with dense image, and $K$ is the composition of the canonical isomorphism $H\cong H^*$ determined by the inner product (i.e. given by Riesz theorem) and of the Banach transpose (dual) $J^*: H^*\to B^*$ of the operator $J$. The fact that $J(B)$ is dense in $H$ implies that $J^*$ (hence also $K$) is injective as well.

\hypertarget{basic_examples}{}\subsection*{{Basic examples}}\label{basic_examples}

A typical example is $B = \mathcal{S}(\mathbb{R}^n)$ (Schwarz space), $H = L^2(\mathbb{R}^n)$ and $B^* = \mathcal{S}'(\mathbb{R}^n)$ (the space of tempered (Schwarz) distributions). One of the basic facts on Fourier transform is that this Gel'fand triple is preserved by the Fourier transform.

Another natural example is $\mathcal{l}^1\hookrightarrow \mathcal{l}^2\hookrightarrow \mathcal{l}^\infty$.

\hypertarget{a_long_example}{}\paragraph*{{A long example}}\label{a_long_example}

Let $H$ be the Hilbert space $L^2(\mathbb{R}, d x)$ consisting of square integrable functions $f$ with respect to [[Lebesgue measure]]. There is an unbounded self-adjoint operator

\begin{displaymath}
m_x: H \to H: f \mapsto x \cdot f
\end{displaymath}
where $(x \cdot f)(y) := y f(y)$. This operator is not defined on all of $H$, but it is defined on a dense subspace of $H$. For example, if $S$ is the Schwartz space consisting of smooth functions $f$ on $\mathbb{R}$ all of whose derivatives $f^{(n)}(x)$ decay rapidly at infinity (more rapidly than any negative power of $|x|$), then there is a dense inclusion map $i: S \to H$, and $m_x$ is defined globally on $S$.

Meanwhile the Schwartz space $S$ carries its own topology (as described in the article [[distribution]]), stronger than the topology it inherits from $H$, and the space of tempered distributions $S^*$ is defined to be the continuous dual of the [[topological vector space|TVS]] $S$. Since the continuous inclusion $i: S \to H$ is dense, it follows that any continuous functional

\begin{displaymath}
f: S \to \mathbb{C}
\end{displaymath}
has at most one extension to a continuous functional $H \to \mathbb{C}$. In other words, the adjoint map

\begin{displaymath}
i^*: H^* \to S^*
\end{displaymath}
is injective. In addition, the topology on $S$ is such that the operator $m_x: S \to S$ is continuous.

In this example, there is a dense inclusion $S \to S^*$ defined by the inner product pairing, and the operator $m_x$ extends uniquely to an operator $S^* \to S^*$, called $m_x$ by abuse of notation. Again, in this example, the operator $m_x: S^* \to S^*$ has an eigenvector $s_{\xi}$ for each $\xi \in \mathbb{R}$:

\begin{displaymath}
m_x(s_{\xi}) = \xi s_{\xi}
\end{displaymath}
(under construction)

\hypertarget{isomorphisms}{}\subsection*{{Isomorphisms}}\label{isomorphisms}

An isomorphism of Gelfand triples $(B_1,H_1,B^*_1)\to (B_2,H_2,B^*_2)$ is a unitary isomorphism $H_1\to H_2$ which restricts to an isomorphism of Banach spaces $B_1\to B_2$, and which extends to a weak$*$- and norm-preserving continuous isomorphism $B_1^*\to B_2^*$.

\hypertarget{terminology_and_references}{}\subsection*{{Terminology and references}}\label{terminology_and_references}

\hypertarget{terminological_variants_and_history}{}\paragraph*{{Terminological variants and history}}\label{terminological_variants_and_history}

Usually, $B$ and $B^*$ are Banach spaces, when we say \emph{Banach Gel'fand triple}, there are some other variants involving more general [[topological vector space]]s. In some cases one also uses the terminology \emph{rigged Hilbert space}, following articles by Roberts and others since mid 1960-s. Nuclear Gel'fand triples are very common and then the notion is well behaved. In Russian literature the term \emph{enriched} Hilbert space is used (  ), sometimes translated also as \emph{equipped} Hilbert space. The enriched word here is the same as in the phrase [[enriched category]].

\hypertarget{early_references}{}\paragraph*{{Early references}}\label{early_references}

Gel'fand triples were introduced by Gel'fand school about 1955 and quickly incorporated into the theory of generalized functions.

\begin{itemize}%
\item [[I. M. Gelfand, A. G. Kostyuenko, \emph{Expansion in eigenfunctions of differential and other operators) (Russian), Dokl. Akad. Nauk SSSR (N.S.) \textbf{103} (1955), 349--352, \href{http://www.ams.org/mathscinet-getitem?mr=73136}{MR73136}}


\item [[I. M. Gel'fand]], N. Ja. Vilenkin, \emph{Generalized functions}, vol. 4. Some applications of harmonic analysis. Equipped Hilbert spaces, Fizmatgiz, Moscow, 1961 \href{http://www.ams.org/mathscinet-getitem?mr=146653}{MR146653}, English transl. Acad. Press 1964 \href{http://www.ams.org/mathscinet-getitem?mr=173945}{MR173945}



\end{itemize}
Related early works include

\begin{itemize}%
\item Ciprian Foia, \emph{D\'e{}compositions int\'e{}grales des familles spectrales et semi-spectrales en op\'e{}rateurs qui sortent de l'espace hilbertien}, Acta Sci. Math. Szeged \textbf{20}, 1959, 117---155. \href{http://www.ams.org/mathscinet-getitem?mr=115092}{MR115092}

\end{itemize}
John Roberts continued the study in the context of [[quantum mechanics]] (in his thesis work suggested by [[Paul Dirac]]), changing the name to rigged.

\begin{itemize}%
\item [[John Roberts]], \emph{The Dirac bra and ket formalism}, J. Mathematical Phys. 7 (1966), 1097--1104, \href{http://www.ams.org/mathscinet-getitem?mr=216836}{MR216836}, \emph{Rigged Hilbert spaces in quantum mechanics} , Comm. Math. Physics \textbf{3}, n. 2 (1966), 98--119, \href{http://www.ams.org/mathscinet-getitem?mr=228243}{MR228243}, \href{http://projecteuclid.org/euclid.cmp/1103839388}{euclid}

\end{itemize}
\hypertarget{modern_references_and_links}{}\paragraph*{{Modern references and links}}\label{modern_references_and_links}

\begin{itemize}%
\item [[rigged Hilbert space]]


\item wikipedia \href{http://en.wikipedia.org/wiki/Rigged_Hilbert_space}{rigged Hilbert space}


\item . . , \href{http://dic.academic.ru/dic.nsf/enc_mathematics/3760/%D0%9E%D0%A1%D0%9D%D0%90%D0%A9%D0%95%D0%9D%D0%9D%D0%9E%D0%95}{  }, online article from (Soviet) Matem. enc.


\item MathOverflow question: \href{http://mathoverflow.net/questions/43313/good-references-for-rigged-hilbert-spaces}{good-references-for-rigged-hilbert-spaces}


\item Feichtinger, \emph{A Banach Gelfand triple framework for regularization and approximation}, \href{http://www.univie.ac.at/nuhag-php/dateien/talks/1086_eucetifeislides.pdf}{slides pdf}



\end{itemize}
A unification of various inequivalent approaches is claimed in

\begin{itemize}%
\item M. Gadella, F. G\'o{}mez, \emph{A unified mathematical formalism for the Dirac formulation of quantum mechanics} Foundations of Physics \textbf{32}, No. 6, (2002)


\item S. Wickramasekara, A. Bohm, \emph{Symmetry representations in the rigged Hilbert space formulation of quantum mechanics}, \href{http://arxiv.org/abs/math-ph/0302018}{math-ph/0302018}



\end{itemize}
[[!redirects Gelfand triple]] [[!redirects Gel'fand triple]] [[!redirects enriched Hilbert space]] [[!redirects equipped Hilbert space]]



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