Let $\Phi=\bigcup_{i\in \mathbb{N}}\Phi_i$ be the inductive limit of an upwardly directed set of countably-seminormed spaces (i.e. the locally convex topology is given by a countable family of separating seminorms $p_k^{\Phi_i}$, which also implies that $\Phi_i$ are Frechet-Spaces), i.e. $$\Phi_1 \subset \Phi_2 \subset \dots$$ $$\tau_1 \subset \tau_2 \subset \dots$$
The topology of $\Phi$ is by definition inductive topollogy, i.e. the strongest locally convex topology, in which the embeddings $\Phi_i \rightarrow \Phi$ are continuous.
I conjecture, that the inductive limit is also a Frechet space. I am trying to construct an explicit form of the family of seminorms on $\Phi$, depending on the norms of the $\Phi_i$ , my first guess is $$ p^\Phi_k(\varphi)=\min_{j\in \mathbb N}\max_{i\leq k}p^{\Phi_i}_j(\varphi) $$ or $p_k^\Phi(\varphi)=0$ if $\varphi \notin \bigcup_{i\leq k}\Phi_i$
By construction, the imbeddings would be continuous with respect the topology generated by this seminorms and would turn $\Phi$ into a Frechet Space, however, I am not able to prove that this is actually equivalent to the inductive topology.
My questions therefore are:
- Is my conjecture right?
- If no, can I at least construct the locally convex topology from the seminorms of $\Phi_i$?
- If yes, does anybody have an idea how to prove, that my topology is actually the inductive topology?
- Alternately to 3., does somebody have a reference on how to construct inductive topologies by seminorms? (I already checked Köthe and Schäfer).
I would greatly appreciate any help.