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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:

  1. Is my conjecture right?
  2. If no, can I at least construct the locally convex topology from the seminorms of $\Phi_i$?
  3. If yes, does anybody have an idea how to prove, that my topology is actually the inductive topology?
  4. 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.

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  • $\begingroup$ No, generally, an inductive limit of Fréchet spaces is not metrisable. $\endgroup$ Commented Jul 27, 2014 at 18:47

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As Daniel Fisher wrote, LF-spaces (countable inductive limits of Frechet spaces) are usually not metrizable, in particular, strict LF-spaces (that is, $\Phi_n$ is a closed topological subspace of $\Phi_{n+1}$) are metrizable if and only if there is $n$ such that $\Phi_m=\Phi_n$ for all $m\ge n$ (to prove this use Grothendieck's factorization theorem).

The description of all continuous seminorms on the limit in terms of the seminorms of the steps is not very handy: You can take either

$\lbrace p$ seminorm on $\Phi$: for all $n$ there are $k$ and $c\ge 0$ such that $p(x)\le c p^{\Phi_n}_k\rbrace$

or the system of seminorms defined for sequences $(c_n)$ and $k(n))$ by

$$ p(x)=\inf \left\lbrace \sum_{n=1}^N c_n p^{\Phi_n}_{k(n)}(x_n): x=\sum_{n=1}^N x_n\right\rbrace.$$

The book Introduction to Funtional Analysis of Meise and Vogt contains some information about locally convex inductive limits.

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