Seminorms generating the topology of the inductive limit of locally-convex spaces

Question

Let $(E_n,\iota_n)$ be an inductive system of LCSs; i.e.: each $\iota_n:E_n\rightarrow E_{n+1}$ is a continuous linear map. Suppose that the topology on each $E_n$ is generated by a (countable) family of semi-norms $\{p_{n,k}:E_n\rightarrow \mathbb{R}\}_{k\in \mathbb{N}}$. I know that the injective limit $\varinjlim_n E_n$ exists in the category of locally-convex spaces (with continuous linear maps as morphisms).

However, I cannot find a description of the semi-norms generating this topology. So my question is, what are the semi-norms generating the LCS-injective limit topology on $\varinjlim_n E_n$?

Answer 1

In the book "barrelled locally convex spaces" by Bonet and Perez Carreras, Proposition 8.4.4 gives another description of a family of seminorms describing the topology of $E_\infty$. With the notation of the above answer, it states that, for all sequences $b_n$ of positive scalars and $k_n$ of positive integers, we get a continuous seminorm $q$ on $E_\infty$ defined by $q(x)=\inf\sum_nb_np_{n,k_n}(x_n)$, where the infimun is taken over all possible representations of $x$ as a finite sum $x=\sum_nx_n$ with $x_n\in E_n$. All possible seminorms $q$ of this type describe the topology of $E_\infty$.

Answer 2

Let me change notations to $\iota_{n+1,n}$ instead of $\iota_n$. Let $E_{\infty}$ denote the inductive limit and let $\iota_{\infty,n}$ be the canonical map $E_n\rightarrow E_{\infty}$. Let $\rho$ be a seminorm on $E_{\infty}$. Call it an admissible seminorm iff for all $n$, $\rho\circ\iota_{\infty,n}$ is a continuous seminorm on $E_n$, i.e., can be bounded by a constant times a maximum of $p_{n,k}$ for finitely many $k$'s. If I remember correctly the topology of the inductive limit $E_{\infty}$ is the one defined by the collection of all admissible seminorms. This is a description "by constraints" as opposed to a parametric description of an explicit collection of seminorms defining the topology. For the particular case of the space of test functions $\mathscr{D}$ see

Doubt in understanding Space $D(\Omega)$