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

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 $$\varinjlim_n E_n$$?

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 $$E_\infty$$.
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 $D(\Omega)$