I have an application where I need to work with the following idea. Let the space $\bigcup_{n=1}^C \mathbb{R}^n$ be associated with the metric $d$ such that for $x=(x_1,\cdots,x_n)$ and $y=(y_1,\cdots,y_m)$, we have $d(x,y) = \begin{cases} \sum_{i=1}^n\vert{x_i-y_i}\vert &\text{if } n=m, \\ \vert{m-n}\vert & \text{Otherwise. } \end{cases}$

A function $\phi$ on $\bigcup_{n=1}^C \mathbb{R}^n$ is a collection of functions $\{\phi_n, 1\leq n\leq C\}$, where $\phi_n$ is defined on $\mathbb{R}^n$.Similarly, let $\nu$ be a finite measure defined on $\bigcup_{n=1}^C \mathbb{R}^n$.

I need to work with the Schwartz space on $\bigcup_{n=1}^C \mathbb R^n$ and its dual by defining an inner product as $\langle\nu,\phi\rangle=\sum_{n=1}^C\int_{\mathbb R^n}\phi(x_1,\dotsc,x_n)\ d\nu(x_1,\dotsc,x_n)$. Do all the properties of Schwartz space and its dual on just $\mathbb R^n$ holds even for the case of the space $\bigcup_{n=1}^C\mathbb R^n$? If it is true, can we extend this to the case when $C=\infty$? Please give some suggestions and references for this.

I work on the stochastic interacting particle systems containing $N$ particles in which each particle state lies in the space $\bigcup_{n=1}^C \mathbb R^n$. The counting measure $\nu$ on $\bigcup_{n=1}^C \mathbb R^n$ can be considered as a collection of $C$ counting measures $\nu=(\nu_n,1\leq n\leq C)$ where $\nu_n$ for $1\leq n\leq C$ is a counting measure on $\mathbb{R}^n$. By considering $\nu$ lies in the dual of a Schwartz space, I want to establish the functional central limit theorem by taking $N\to\infty$ using the Mitoma's theorem. In general, the results are known for the case when $\nu$ lies on just $\mathbb{R}^n$. In my case, I can view $\nu$ as a single measure on $\bigcup_{n=1}^C \mathbb R^n$ or a collection of measures $(\nu_n,1\leq n\leq C)$ where $\nu_n$ lies on $\mathbb{R}^n$. If I view $\nu$ as collection of measures, a sequence of measures $\{\eta^m\}$ converges to $\eta$ if $\{\eta^m_i\}$ converges to $\eta_i$ for all $1\leq i\leq C$. I am looking at the notion of the convergence of the measures on $\bigcup_{n=1}^C \mathbb R^n$ by considering they lie in the dual of a Schwartz space. Please let me know any useful references or suggestions.

needto do that? Taking a setwise union of vector spaces is something so weird that I would like to know what drove you to it. I mean, even the types of the elements don't match. Why would anyone ever want to unionise these spaces? Are you sure you're not trying to do something completely different here? $\endgroup$ – Johannes Hahn Aug 8 '18 at 22:19