A Gel’fand triple is a structure that equips a Hilbert space with a dense 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 eigenvectors for the continuous spectrum of normal (possibly unbounded) linear operators.
A Gel’fand triple is datum of the form
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.
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 $\mathcal{l}^1\hookrightarrow \mathcal{l}^2\hookrightarrow \mathcal{l}^\infty$.
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
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 TVS $S$. Since the continuous inclusion $i: S \to H$ is dense, it follows that any continuous functional
has at most one extension to a continuous functional $H \to \mathbb{C}$. In other words, the adjoint map
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 $\xi \in \mathbb{R}$:
(under construction)
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 $B_1^*\to B_2^*$.
Usually, $B$ and $B^*$ are Banach spaces, when we say Banach Gel’fand triple, there are some other variants involving more general topological vector spaces. In some cases one also uses the terminology 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 enriched Hilbert space is used (оснащенное гильбертово пространство), sometimes translated also as equipped Hilbert space. The enriched word here is the same as in the phrase enriched category.
Gel’fand triples were introduced by Gel’fand school about 1955 and quickly incorporated into the theory of generalized functions.
[[I. M. Gelʹfand, A. G. Kostyučenko, Expansion in eigenfunctions of differential and other operators) (Russian), Dokl. Akad. Nauk SSSR (N.S.) 103 (1955), 349–352, MR73136
I. M. Gel'fand, N. Ja. Vilenkin, Generalized functions, vol. 4. Some applications of harmonic analysis. Equipped Hilbert spaces, Fizmatgiz, Moscow, 1961 MR146653, English transl. Acad. Press 1964 MR173945
Related early works include
John Roberts continued the study in the context of quantum mechanics (in his thesis work suggested by Paul Dirac), changing the name to rigged.
wikipedia rigged Hilbert space
Р. А. Минлос, Оснащенное гильбертово пространство, online article from (Soviet) Matem. enc.
MathOverflow question: good-references-for-rigged-hilbert-spaces
Feichtinger, A Banach Gelfand triple framework for regularization and approximation, slides pdf
A unification of various inequivalent approaches is claimed in
M. Gadella, F. Gómez, A unified mathematical formalism for the Dirac formulation of quantum mechanics Foundations of Physics 32, No. 6, (2002)
S. Wickramasekara, A. Bohm, Symmetry representations in the rigged Hilbert space formulation of quantum mechanics, math-ph/0302018