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I am in the process of writing some self-contained notes on probability theory in spaces of distributions, for the purposes of statistical mechanics and quantum field theory. Perhaps the simplest approach is to follow Barry Simon's philosophy in his book "Functional Integration and Quantum Physics", namely, evacuate the complexity of spaces of test functions right from the start and replace them by more concrete spaces of sequences to which they are isomorphic as topological vector spaces. For instance the Schwartz space $S(\mathbb{R}^d)$ is isomorphic to the space $\mathcal{s}$ of rapidly decaying sequences. A very nice proof of this fact using Hermite functions can be found for instance here.

For applications to QFT it is also important to consider the space $S(\mathbb{Q}_p^d)$ of locally constant compactly supported functions over a $p$-adic field. This allows one to work with hierarchical models as a toy model for Euclidean models. This hierarchical simplification is similar to the use of branching random walks or Mandelbrot cascades as a toy model for the massless free field (see for instance this article by Bramson and Zeitouni or this one by Benjamini and Schramm) in probability theory or that of Walsh series as a toy model for Fourier series in harmonic analysis (see for instance this article by Do and Lacey in the hierarchical case and this one by Lie for the Euclidean case). The relevant space $S(\mathbb{Q}_p^d)$ is easily seen to be isomorphic to the space $\mathcal{s}_0=\mathbb{R}^{(\mathbb{N})}$ of almost finite sequences with the finest locally convex topology.

My question concerns the existence of similar isomorphism theorems for more complicated spaces of test functions. Let $\Omega$ be a nonempty open set in $\mathbb{R}^d$ and let $\mathcal{D}(\Omega)$ be the usual space of smooth compactly supported functions with support contained in $\Omega$. Let $S(\mathbb{A}_{\mathbb{Q}})$ and $S(\mathbb{A}_{\mathbb{Q}}^{\times})$ be the spaces of Schwartz-Bruhat functions respectively over the adeles and ideles of $\mathbb{Q}$. Finally let $\mathcal{s}^{(\mathbb{N})}$ be the space of almost finite sequences of elements in $\mathcal{s}$. The latter is equipped with the locally convex topology defined by all semi-norms which are continuous when restricted to each summand $\mathcal{s}$.

I suspect the following is true: $$ \mathcal{D}(\Omega)\simeq S(\mathbb{A}_{\mathbb{Q}}) \simeq S(\mathbb{A}_{\mathbb{Q}}^{\times}) \simeq \mathcal{s}^{(\mathbb{N})} $$ as topological vector spaces.

Question 1: is this true or not and if it is where can I find a proof?

Question 2: is the space $\mathcal{s}^{(\mathbb{N})}$ sequential?


Edit 1: The references in the answer by P. Michor were spot-on regarding the isomorphism $\mathcal{D}(\Omega)\simeq \mathcal{s}^{(\mathbb{N})}$ which now I know is true. However I would prefer a more direct proof (which does not use the result by Pełczyński about spaces being isomorphic to a complemented subspace of the other). An ideal proof for me would be via the use of something like a wavelet basis $\psi_{p,x}$ where $p$ would be the scale index and $x$ the spatial location index. Intuitively, for $\mathcal{D}(\Omega)$ one has to be almost finite in the $x$ direction and suitably decaying in the $p$ direction hence a space of the form $\mathcal{s}\otimes \mathcal{s}_0$. For $S(\mathbb{R}^d)$ one would have decay in both directions resulting in $\mathcal{s}\otimes \mathcal{s}\simeq \mathcal{s}$.

As for the second isomorphism, it is rather clear how to construct a linear isomorphism. So the question really is: what is the "standard" topology on $S(\mathbb{A}_{\mathbb{Q}})$ and $S(\mathbb{A}_{\mathbb{Q}}^{\times})$? Number theory references I looked at tend to dodge the question altogether.


Edit 2: A possible idea to cover the case of general open sets $\Omega$ is to use some diffeomorphisms in order to relate them to $\mathbb{R}^d$. However, this MO question does not bode well for such an approach.

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Check the following sources:

  • MR0688001 Reviewed Vogt, Dietmar Sequence space representations of spaces of test functions and distributions. Functional analysis, holomorphy, and approximation theory (Rio de Janeiro, 1979), pp. 405–443, Lecture Notes in Pure and Appl. Math., 83, Dekker, New York, 1983. (Reviewer: M. Valdivia)

  • M. Valdivia: Topics in locally convex spaces, North Holland, 1982.

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A construction for the isomorphism not using the Pełczyński decomposition method can be found in

  • C. Bargetz: Explicit representations of spaces of smooth functions and distributions. J. Math. Anal. Appl. 424: 149–1505. 2015 DOI: 10.1016/j.jmaa.2014.12.009
  • C. Bargetz: Commutativity of the Valdivia–Vogt table of representations of function spaces. Mathematische Nachrichten 287(1): 10–22, 2014, DOI: 10.1002/mana.201200258.

I do not think that the basis constructed in the first of the above papers really has the properties you would like to have.

The construction of the isomorphism on the other hand consists of a series of relatively simple steps and it is completely constructive. More precisely, in the one dimensional case, the construction works as follows: First, using Seeley's extension operator, the function is "split" into a sequence of smooth functions on the unit interval $[0,1]$ which together with all derivatives vanish at the right boundary point $1$. Then, using an explicit formula each of these smooth functions is mapped to an element of $s$.

Some topological properties of $s^{(\mathbb{N})}$ and related spaces are dicussed in

  • C. Bargetz: Completing the Valdivia–Vogt tables of sequence-space representations of spaces of smooth functions and distributions. Monatsh. Math. 177(1): 1–14, 2015. DOI: 10.1007/s00605-014-0650-2
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