# Are countable strict inductive limits of Fréchet spaces always LF-spaces?

I would like to work with a slightly loser definition of an LF-space but am unsure what niceties I'm throwing away in the process. Let me provide a comparison of the conventional definition and my own slightly broader one.

Generalized LF-Topology: Let $E$ be a vector space over $K = \mathbb{R}$ or $\mathbb{C}$. Let $\{E_i : i \in I\}$ be a countable family of $K$-linear subspaces of $E$ such that

1. The relation $i \leq j \iff E_i \subseteq E_j$ defines a partial order on $I$.
2. We have $$E = \varinjlim_{i \in I}E_i = \bigoplus_{i \in I} E_i \bigg/ \langle g_i - g_j \circ g_{j,i} \mid i \leq j\rangle,$$ where for each $i \in I$ we define the canonical map $g_i: E_i \to \oplus_{j \in I}E_j$ and for each $j \geq i$ we define the inclusion map $g_{j,i}: E_i \to E_j$. (Here the topology on $\oplus_{i \in I}E_i$ is defined to be the finest locally convex topology such that $g_i$ is continuous for all $i \in I$.
3. For each $i \in I$ we have fixed a topology $\tau_i$ on $E_i$ with respect to which $E_i$ is a Fréchet space.
4. The directed family of LCSs $\{(E_i,\tau_i) : i \in I\}$ is strict, meaning that if $i \leq j$ the topology on $E_i$ induced by $\tau_j$ is exactly $\tau_i$. (If we only required that the imbedding $E_i \to E_j$ was continuous, then the induced topology could be coarser than $\tau_i$.

After potentially adding more restrictions to $(I,\leq)$ and the $E_i$, the LF-topology on $E$ may be defined as follows: A convex subset $U$ of $E$ is an open neighborhood of $0$ in $E$ if and only if $U \cap E_i$ is open in $E_i$ for all $i \in I$.

Conventional Definition: My definition above, but where it is assumed that $I = \mathbb N$ under the usual total order.

My question is: if I permit partially ordered countable families of Fréchet spaces, but still require the inductive limit to be strict (as defined above), do I wind up with the same class of LF-spaces, or is this definition actually more general? Are there any modern references (say, post early 1970's) that have studied broader definitions of LF-spaces than those that were investigated classically by Dieudonné-Schwartz and Köthe?

Edit: Thinking about this a bit more, it seems at least convenient (and probably necessary) to assume further that the poset $I$ is well-founded.

• If the family is directed in the sense that for all $i,j$ there is a $k$ with $E_i \cup E_j \subseteq E_k$, then you get nothing new. If there are $i,j$ such that $(E_i \cup E_j)\setminus E_k \neq \varnothing$ for all $k$, I'm not sure whether $\bigcup E_k$ can be a vector space, maybe you then need more than countably many spaces to get a vector space as the union. – Daniel Fischer Jun 13 '15 at 21:27
• @DanielFischer: Whoops! I was being too sloppy with my assumptions in the definition. I was thinking about the more general case you mention, but the definition of the limit as a union doesn't make sense here. Thanks for the reply! I'll update my question. – Dan Jun 13 '15 at 21:29

Fact: The limit $E$ of a countable inductive system $(E_i)_{i \in I}$ of Fréchet spaces is already an (LF)-space in the conventional sense, i.e. $E$ is the inductive limit of a sequence of Fréchet spaces $(E_n)_{n \in \mathbb{N}}$.
Thus, you do not get anything new when considering general countable directed index sets $I$. You do not need the system to be strict. For a reference see Meise, Vogt: "Introduction to Functional Analysis" (2004), p. 291 (Remark). The construction of the sequence can be found in Lemma 24.34: Forget about the partial order on $I$ (induced by $\subseteq$ on $E$) and identify $I = \mathbb{N}$ (hereby inducing a linear order on $I$). Then the sequence $(E_n)_{n \in \mathbb{N}}$ has an increasing subsequence $(E_{n_k})_{k \in \mathbb{N}}$ (with $E_\nu \subseteq E_{n_{k+1}}$ for all $1 \leq \nu \leq n_k$ and all $k$) which produces the same inductive limit topology on $E$:
$$\displaystyle \lim_{\stackrel{\longrightarrow}{i \in I}} E_i = \lim_{\stackrel{\longrightarrow}{k \in \mathbb{N}}} E_{n_k}.$$