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.

  • $\begingroup$ No, generally, an inductive limit of Fréchet spaces is not metrisable. $\endgroup$ – Daniel Fischer Jul 27 '14 at 18:47

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.