Let $S(\mathbb{N})$ be the space of rapidly decreasing sequences and $S'(\mathbb{N})$ its topological dual, the space of sequences bounded by a polynomial.
For $m\in \mathbb{Z}$, we also define $\ell\_2^m (\mathbb{N})$ as Hilbert spaces of sequences such that $(u\_n (n+1)^m)\_{n\in \mathbb{N}} \in \ell\_2 (\mathbb{N})$.


It is known than $S(\mathbb{N})$ is the projective limit of the spaces $\ell\_2^m (\mathbb{N})$.
As such, $S(\mathbb{N})$ is a Frechet space with a nuclear topology.
Its dual $S'(\mathbb{N})$ is hence the inductive limit of the spaces $(\ell\_2^m)' (\mathbb{N}) = \ell\_2^{-m} (\mathbb{N})$. So:


**Fact 1:** $S'(\mathbb{N})$ has a natural complete nuclear topology defined as a countable inductive limit of Hilbert spaces.


It is also known that any complete nuclear space is isomorphic with the projective limit of a suitable family of Hilbert spaces. See for instance Corollary 3, Section 7.2 of [Topological Vector Spaces](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=JOUR&pg8=ET&review_format=html&s4=Schaefer&s5=Topological%20vector%20spaces&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=1&mx-pid=1741419).


**Fact 2:** $S'(\mathbb{N})$ has an abstract complete nuclear topology as a projective limit of Hilbert spaces.


**Question:** Is it possible to describe the topology of $S'(\mathbb{N})$ as a countable projective limit of Hilbert spaces $H\_m$, meaning that $S'(\mathbb{N}) =\bigcap\_{m\in \mathbb{N}} H\_m$, such that the $H\_m$ are described as sequence spaces (bigger than $S'(\mathbb{N})$ of course)?



Is the space $S'(\mathbb{N})$ of slowly increasing sequences the projective limit of Hilbert sequence spaces?
I can now answer my own question thanks to the following discussion: [Which Fréchet spaces have a dual that is a Fréchet space?](https://mathoverflow.net/questions/63383/which-fr%C3%A9chet-spaces-have-a-dual-that-is-a-fr%C3%A9chet-space)


* As explained in the comments, $\mathcal{S}'(\mathbb{N})$ can be defined as the projective limit of a family of semi-norms $(p\_u)$ indexed by $u \in \mathcal{S}(\mathbb{N})$.
* It is however impossible to extract a countable family from the $(p\_u)$ due to the much more general fact that the strong dual of a Frechet space $E$ is metrizable if and only if $E$ is normable. Here, $E = \mathcal{S}(\mathbb{N})$ is not normable, so $E' = \mathcal{S}'(\mathbb{N})$ is not metrizable, hence cannot be defined as a projetive limit of a countable family of sequence spaces.


