Before I go into the question, I would like to mention that I am a physicist with some formal mathematical knowledge, but not expert in functional analysis.

In physics, we very often say: Let $|x \rangle$ be the position basis, let $| p \rangle$ be the momentum basis. We then proceed to say $\langle x| x' \rangle=i \hbar \delta(x-x')$ and similarly up to some factor for the momentum. They are related by the Fourier transform $\langle x |p \rangle=e^{ipx}$. Strictly speaking, this cannot be correct, as the "eigenfunctions" of the position and momentum operator are not elements of the Hilbert space, they are not square integrable. One could say, the set of eigenvectors in $H$ is empty.

I would like to give this a precise mathematical meaning and relate this to the spectrum. I am fully aware of the very fact which is pointed out in mathematical physics texts, such as Simon-Reed, or Moretti, that what we ultimately measure is the spectrum and that can be read off simply from the spectral theorem without any referrence to eigenvectors. So strictly speaking to do this, is not a necessity, and QM is complete even without this. However, if physicists use this, I am confident that this can be made precise! :)

After long search, I found out that the language physicists use can be made precise in the Rigged Hilbert space setting. This precisely means that we choose a dense subspace of $H$ and take its topological dual, and embed $H$ into the dual. Then we can make sense of "position and momentum eigenkets" as elements of this topological dual. Now, what I would like to see:

1. Let $A$ be a self-adjoint (possibly unbounded) operator on $H$. Then corresponding to the purely continuous spectrum, there are no eigenvectors of $A$ in $H$, but there exist generalized eigenvectors of $A$ in the topological dual of the dense subspace.
2. the spectral theorem in this nuclear setting (theorem 1 in the link), or in more concrete terms: Let $A$ be a self-adjoint operator on an infinite dimensional separable Hilbert space $H$ with simple absolutely continuous spectrum $\sigma(A)=\mathbb{R}^{+}$. Then, there exists a Rigged Hilbert space $\phi \subset H \subset \phi^{\times}$, such that

a) $A \phi \subset \phi$ and $A$ is continuous on $\phi$. Therefore it may be continuously extented to $\phi^{\times}$.

b) For almost all $\omega \in \mathbb{R}^{+}$, with respect to the Lebesgue measure, there exists $| \omega \rangle \in \phi^{\times}$ so that $A | \omega \rangle=\omega | \omega \rangle$.

c) For any pair of vectors $\varphi,\psi \in \phi$, and any measurable function $f:\mathbb{R}^{+} \to \mathbb{C}$, we have that $$\langle \varphi| f(A) \psi \rangle= \int_{0}^{\infty} f(\omega) \langle \varphi | \omega \rangle \langle \omega| \psi \rangle d \omega,$$ with $\langle \omega|\psi \rangle^{*}=\langle \psi | \omega \rangle$.

d) The above spectral decomposition is implemented by a unitary operator $U:H \to L^2(\mathbb{R}^{+})$, with $U (\psi)=\langle \omega| \psi \rangle=\psi(\omega)$ and $(U A U)^{-1} \psi(\omega)=\omega \psi(\omega)=\omega \langle \omega| \psi \rangle$ for any $\psi \in \phi$. This means that $UAU^{-1}$ is the multiplication operator on $U \psi$.

e) For any pre existent Rigged Hilbert space $\phi \subset H \subset \phi^{\times}$, such that $A \phi \subset \phi$ with continuity and $A$ is satisfying our hypotheses, then items $b)-d)$ hold.

I haven't seen a proof of 1), but the statement can be found in Frederic Schuller's lecture notes, page 191, proposition 20.2.

For the second, 2) I have seen four proofs in the literature, which partly prove it, but none fully and none of them is followable/student-friendly. One is in Blanchard's mathematical physics book, and the second proof is in Maurin's book, the third being in an article by Gould. The fourth proof (original by Maurin,Gelfand) seems to be incorrect, and is pointed out in the translation of the book.

So my question is: can anyone recommend student-friendly sources (say, starting from functional analysis as covered by Moretti's book on Spectral theory and QM, or Frederic Schuller's lectures), which prove these two theorems? I am aware that this might probably be hard to find, but in case there is no other literature than I provided, can perhaps some expert in the field direct me to what prerequisites should I learn/where can I find the material that I need to write myself a self-contained proof following one of the proofs in the literature?

P.S. According to the nuclear theorem part e), is the Schwartz space subset Hilbert space subset space of tempered distributions the unique up to unitary equivalence of Gelfand triples Rigged Hilbert space, which makes the construction in physics work?