Every now and then I attempt to understand better quantum mechanics and quantum field theory, but for a variety of possible reasons, I find it very difficult to read any kind of physicist account, even when the physicist is trying to be mathematically respectable. (I am not trying to be disrespectful or controversial here; take this as a confession of stupidity if it helps.) I am generally interested in finding online mathematical accounts which ideally would come close to being of "Bourbaki standard": definition-theorem-proof and written for mathematicians who prefer conceptual explanations, and ideally with tidy or economical notation (e.g., eschewing thickets of subscripts and superscripts).
More specifically, right now I would like a (mathematically trustworthy) online account of rigged Hilbert spaces, if one exists.
Maybe I am wrong, but the Wikipedia account looks a little bit suspect to me: they describe a rigged Hilbert space as consisting of a pair of inclusions $i: S \to H$, $j: H \to S^\ast$ of topological vector space inclusions, where $S^\ast$ is the strong dual of $S$, $H$ is a (separable) Hilbert space, $i$ is dense, and $j$ is the conjugate linear isomorphism $H \simeq H^\ast$ followed by the adjoint $i^\ast: H^\ast \to S^\ast$. This seems a little vague to me; should $S$ be more specifically a nuclear space or something? My guess is that a typical application would be where $S$ is Schwartz space on $\mathbb{R}^4$, with its standard dense inclusion in $L^2(\mathbb{R}^4)$, so $S^\ast$ consists of tempered distributions.
I also hear talk of a nuclear spectral theorem (due to Gelfand and Vilenkin) used to help justify the rigged Hilbert space technology, but I don't see precise details easily available online.
Best Answer
Some time ago I was interested in rigged Hilbert space to get a better understanding of quantum physics. On that occasion I collected some references on this subject, see below. It's quite comprehensive. A good starting point for an overview could be the works of Madrid and Gadella. Note that there are different versions of "rigged Hilbert space" (in context of quantum physics) in literature.
J.-P. Antoine. Dirac formalism and symmetry problems in quantum mechanics. i. general dirac formalism. Journal of Mathematical Physics, 10(1):53--69, 1969.
N.Bogoliubov, A.Logunov, and I.Todorov. Introduction to Axiomatic Quantum Field Theory, chapter 1 Some Basic Concepts of Functional Analysis 4 The Space of States, pages 12--43, 113--128. Benjamin, Reading, Massachusetts, 1975.
R.de la Madrid. Quantum Mechanics in Rigged Hilbert Space Language. PhD thesis, Depertamento de Fisica Teorica Facultad de Ciencias. Universidad de Valladolid, 2001. (available here) The given link is broken: this one is OK http://arxiv.org/pdf/quant-ph/0502053.pdf (Tom Collinge 25 June 2016)
M.Gadella and F.Gómez. A unified mathematical formalism for the dirac formulation of quantum mechanics. Foundations of Physics, 32:815--869, 2002. (available here)
M.Gadella and F.Gómez. On the mathematical basis of the dirac formulation of quantum mechanics. International Journal of Theoretical Physics, 42:2225--2254, 2003.
M.Gadella and F.Gómez. Dirac formulation of quantum mechanics: Recent and new results. Reports on Mathematical Physics, 59:127--143, 2007.
I.M. Gelfand and N.J. Vilenkin. Generalized Functions, vol. 4: Some Applications of Harmonic Analysis, volume4, chapter 2-4, pages 26--133. Academic Press, New York, 1964.
A.R. Marlow. Unified dirac-von neumann formulation of quantum mechanics. i. mathematical theory. Journal of Mathematical Physics, 6:919--927, 1965.
E.Prugovecki. The bra and ket formalism in extended hilbert space. J. Math. Phys., 14:1410--1422, 1973.
J.E. Roberts. The dirac bra and ket formalism. Journal of Mathematical Physics, 7(6):1097--1104, 1966.
J.E. Roberts. Rigged hilbert spaces in quantum mechanics. Commun. math. Phys., 3:98--119, 1966. (available here)
Tjøstheim. A note on the unified dirac-von neumann formulation of quantum mechanics. Journal of Mathematical Physics, 16(4):766--767, 4 1975.
Edit I remember that there is also a discussion about Gelfand triples in physics in the Funktionalanalysis books by Siegfried Großmann but I don't have a copy handy the moment. Though it is in german it might be interesting for you, too.