Let $V=\mathbb{R}^n$, $\Lambda_r=2\pi r \mathbb{Z}^n \subset V (r>0)$ a lattice; $V^*\cong\mathbb{R}^n$ the dual vector space of $V$, and $\Lambda_r^*=\frac{1}{2\pi r} \mathbb{Z}^n =\text{Hom}(\Lambda_r, \mathbb{Z})$ the dual lattice in $V^*$.

$\Lambda_r^*$ can be thought of as the [Pontryagin dual][0] of the torus $T^n_r=V/\Lambda_r$; also, $V^*$ can be thought of as the Pontryagin dual of $V$ and can be identified with $V$ via the pairing $\left< x,\xi \right>=e^{2\pi i x\cdot\xi}$. Chapter 4 of Gerald B. Folland's book [A course in abstract harmonic analysis][1] is a nice introduction to these material in the context of locally compact abelian groups; see also [this blog][2] of Terence Tao.

It's well known that the Fourier transform gives an isometry of Hilbert spaces
$$L^2(V)\cong L^2(V^*).$$

Also, Fourier series give an isometry of Hilbert spaces
$$L^2(T^n_r)\cong l^2(\Lambda_r^*).$$

We have the following obvious intuition: as $r>0$ becomes larger and larger, the scale of $T^n_r$ also becomes larger and larger, and finally becomes like $V=\mathbb{R}^n$; on the other hand, the dual lattice $\Lambda_r^*$ becomes more and more 'dense' in $V^*=\mathbb{R}^n$ as the distance of adjacent points is $\frac{1}{2\pi r}$, which goes to 0 as $r$ goes to $\infty$.

My question is the following:
>Can we make it mathematically rigorous, both on the level of functions and on the level of spaces (e.g. $T^n_r \to V$), that the 'limit' of the isomorphisms
$$L^2(T^n_r)\cong l^2(\Lambda_r^*)$$
is the isomorphism
$$L^2(V)\cong L^2(V^*)$$
as $r$ goes to $\infty$?

The bad thing is that $V=\mathbb{R}^n$ is noncompact, while we have the notion of [Bohr compactification][3], I hope this can be helpful.
>Is there any relation between the tori $T^n_r$ and the Bohr compactification of $\mathbb{R}^n$?

Hopefully, if we can do this, then we can do similar thing such as interpreting Fourier inversion as a limit.

[0]: https://en.wikipedia.org/wiki/Pontryagin_duality
[1]: https://books.google.com.hk/books?id=z-GYCgAAQBAJ&printsec=frontcover&dq=a+course+in+abstract+harmonic+analysis+second+edition+chapter+4&hl=en&sa=X&redir_esc=y#v=onepage&q=a%20course%20in%20abstract%20harmonic%20analysis%20second%20edition%20chapter%204&f=false
[2]: https://terrytao.wordpress.com/2009/04/06/the-fourier-transform/
[3]: https://en.wikipedia.org/wiki/Bohr_compactification