We denote by $\mathcal{S}(\mathbb{R})$ the space of smooth and rapidly decreasing functions. We define on $\mathcal{S}(\mathbb{R})$ the family of semi-norms 
$$\lVert \varphi \lVert\_{n,m} = \lVert (1+|\cdot|^m) \varphi^{(n)} \lVert\_\infty.$$
This family of semi-norms defines a topology $\tau$ on $\mathcal{S}(\mathbb{R})$. The space $(\mathcal{S}(\mathbb{R}),\tau)$ is then well-known to be nuclear.


If $\mathcal{M}(\mathbb{R})$ is the space of measurable functions in $\mathbb{R}$, I define the space $\mathcal{R}(\mathbb{R})$ of rapidly decreasing functions (without smoothness condition) by 
$$\mathcal{R}(\mathbb{R}) = \{ \varphi \in \mathcal{M}(\mathbb{R}), \ \lVert \varphi \lVert\_{0,m} < \infty \}.$$
Moreover, the family of (semi-)norms $(\lVert \cdot \lVert\_{0,m})\_{m\in \mathbb{N}}$ defines a topology $\tau'$ on $\mathcal{R}(\mathbb{R})$. 


**Question: Is $(\mathcal{R}(\mathbb{R}),\tau')$ a nuclear space?**


NB. The proofs I found for the nuclearity of $\mathcal{S}(\mathbb{R})$ are essentially using the facts that the space $s$ of sequences with quick decay is nuclear and $\mathcal{S}(\mathbb{R})$ is homeomorphic to $s$. 


Several authors are claiming that the nuclear structure is strongly related with the condition of infinite smoothness. The space $\mathcal{D}(\mathbb{R})$ of compactly supported and smooth function (with the inductive topology coming from the spaces $\mathcal{D}([-n,n])$), and the space $\mathcal{C}^\infty(\mathbb{R})$ (projective limit of the spaces $\mathcal{D}([-n,n])$) are other famous examples of nuclear structure that go in that sense. This suggests that the space $\mathcal{R}(\mathbb{R})$ may be not nuclear or at least that the usual technics are not available.



Is the space of rapidly decreasing (non-smooth) functions nuclear?
Your space contains an isomorphic copy of $L^\infty([0,1])$ (consider the family of elements with support in the interval) and so is not nuclear.


By the way an explanation of the relationship between smoothness and nuclearity is that many spaces of test functions are generated in a natural way by differential 
operators as described in a short but important article by A. Pietsch (Math. Ann. vol. 164). This article contains a simple criterion for such spaces to be nuclear, based on the spectral properties of the operators as unbounded, self-adjoint ones on Hilbert space and displays some of the standard spaces as explicit examples.


