TITLE: Is the space $S'(\mathbb{N})$ of slowly increasing sequences the projective limit of Hilbert sequence spaces?
QUESTION [8 upvotes]: 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.
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)?

REPLY [5 votes]: I can now answer my own question thanks to the following discussion: Which Fréchet spaces have a dual that is a Fréchet 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.