Good references for Rigged Hilbert spaces? 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.
 A: I would highly recommend looking at the chapter on Sobolev Towers in the book by Engel and Nagel One-Parameter Semigroups for Linear Evolution Equations or the "baby edition" A Short Course on Operator Semigroups.
It provides a really nice example of rigged Hilbert spaces. For example, if $A:D(A) \subset L^2 \to L^2$ is the (Dirichlet) Laplacian, then one can identify $D(A^n)$, $n=1,2,\ldots$ with Sobolev spaces and $D(A^{-n})$ with the negative Sobolev spaces (i.e. extrapolation spaces of $A$).
This concept can be taken further if one considers analytic semigroups and fractional powers of operators and also into the Banach space setting (see Amann's book Linear and Quasilinear Parabolic Problems: Abstract linear theory).
Basically, the concept of rigged Hilbert spaces becomes really natural if one keeps PDEs and Sobolev spaces in mind.
Finally, the book by Reed and Simon Methods of Modern Mathematical Physics - Vol 1: Functional analysis provides a number of references for rigged Hilbert spaces  at the end of Section VII (page 244).
A: "Generalized functions volume 4" by Gelʹfand, Vilenkin, (Math review number 0146653) has a long an detailed discussion of rigged Hilbert spaces and nuclear spaces. The book by Glimm and Jaffe  has a brief summary of the theory. 
A: This is not precisely related to your question, but a certain notion of rigged Hilbert space occurs in the theory of C*-algebras. Particularly, one should look at the work of Marc Rieffel, e.g. https://math.berkeley.edu/~rieffel/papers/morita_equivalence.pdf. I figured I'd mention this because it is decidedly mathematical, and a useful idea.
A: The Springer online Encyclopedia of Mathematics' entry on RHS looks more rigorous albeit also more succinct than Wikipedia; for another online intro see the nlab entry. In addition to the references listed there, a rigorous discussion of the RHS can be found (as far as I recall -- I do not have a copy handy) e.g. in the two-volume book Principles of Advanced Mathematical Physics by Robert D. Richtmyer. Also, it appears that, unlike the physics community, the name Gelfand triple (rather than RHS) is more commonly used by the mathematicians.
A: I would add A.Bohm and M.Gadella, Dirac Kets, Gamow Vectors and Gel'fand Triplets, Springer Lecture Notes in Physics, vol. 348 (Springer, Berlin, 1989) https://doi.org/10.1007/3-540-51916-5
A: The truth is that an actual rigorous theory of rigged Hilbert space doesn't exist at all. That's the bitter truth. Unfortunately we have been told for many years to don't worry, that this theory was existing, somewhere. However it looks like this is a mythology....I did many investigations for many years.
