Do Hausdorff locally convex inductive limits always exist?

Question

The following is from Schaefer, "Topological Vector Spaces", 1999, p. 56/57:

Let $(E_\alpha)_{\alpha \in A}$ be a family of locally convex spaces with $\alpha$ in a directed poset $A$ and $h_{\beta \alpha} : E_\alpha \to E_\beta$ continuous linear maps for $\alpha \leq \beta$. Set $F := \oplus_\alpha E_\alpha$ and let $g_\alpha : E_\alpha \to F$ be the canonical imbedding. Denote by $H$ the subspace of $F$ generated by the ranges of all the maps $g_\alpha - g_\beta h_{\beta \alpha} : E_\alpha \to F$ for $\alpha \leq \beta$.

If $H \subseteq F$ is closed then one can represent the inductive limit of the family $E_\alpha$ in the category of Hausdorff locally convex spaces by $F / H$ which is then a Hausdorff locally convex space.

Now Schaefer writes: "It appears to be unknown whether $H$ is necessarily closed in $F$."

It seems that this statement is equivalent to say that the inductive limit does not necessarily exist.

Does one know here whether this is still an open issue?

EDIT: Summarizing two answers: The inductive limit $F / H$ in the cat. of l.c.s. does exist and is not necessarily Hausdorff. Also, the inductive limit in the cat. of Hausdorff l.c.s. does also exist and is equal to $F / \overline{H}$. For closed $H$ they coincide.

Answer 1

It is quite well-known that locally convex inductive limits need not be Hausdorff.

There are examples of non-Hausdorff locally convex inductive limits of nuclear Fréchet spaces in Klaus Floret's article Some aspects of the theory of locally convex inductive limits, Functional analysis: surveys and recent results, II (Proc. Second Conf. Functional Anal., Univ. Paderborn, Paderborn, 1979), pp. 205–237, North-Holland Math. Stud., 38, North-Holland, Amsterdam-New York, 1980.

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Edit. A former version of this example was flawed as was spotted by Pietro Majer. I apologize that his insightful comments are no longer relevant. Moreover I thank Leonhard Frerick for the discussion yielding this (hopefully) improved version.

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Here is my favorite example. It is the (countable and injective) limit of nuclear Frechet spaces.

Let $U_n\subseteq\mathbb C$ be the balls with center $1/n$ and radius $1/n$ and set $X_n=H(U_n)$, the space of holomorphic functions on $U_n$ endowed with the topology of uniform convergence on compact subsets ($X_n$ is thus a nuclear Frechet space). Since the restriction $X_{n}\to X_{n+1}$ $f\mapsto f|_{U_{n+1}}$ is injective we may consider $X_n$ as a subspace of $X_{n+1}$ with continuous inclusion. Then $X_\infty=$ind $X_n$ has the trivial topology.

By the Hahn-Banach theorem we have to show that every continuous linear functional $\varphi$ on $X_\infty$ is zero.

Since the restriction of $\varphi$ to $X_1$ is continuous there are a convex compact subset $K_1$ of $U_1$ and $c_1>0$ with $|\varphi(f)|\le c_1\|f\|_{K_1}$ for all $f\in X_1$ (where $\|\cdot\|_K=\sup\{|f(z)|:z\in K\}$). We can then choose $n_0\in\mathbb N$ such that the closure of $U_{n_0}$ is disjoint from $K_1$, and we will show $\varphi(g)=0$ for all $g\in X_n$ with $n\ge n_0$ (which thus implies $\varphi=0$, as desired).

Again by the continuity of $\varphi$, there are a convex compact set $K_n\subseteq U_n$ and $c_n>0$ with $|\varphi(g)|\le c_n\|g\|_{K_n}$ for $g\in X_n$.

For an open set $V$ containing $K_1$ with $U_n\cap V=\emptyset$ and $g\in X_n$ we define a holomorphic function $h$ on $U_n\cup V$ by $h(z)=g(z)$ for $z\in U_n$ and $h(z)=0$ for $z\in V$. Since $\mathbb C\setminus (K_1\cup K_n)$ is connected Runge's theorem yields a sequence of entire functions $f_k$ with $f_k\to h$ uniformly on $K_1\cup K_n$. Since $f_k\in X_n$ for all $n\in\mathbb N$ we obtain $$\varphi(g)=\lim_{k\to\infty} \varphi|_{X_n}(f_k)=\lim_{k\to\infty} \varphi|_{X_1}(f_k)=0.$$

Answer 2

The limit always exists whether or not $H$ is closed: The inductive limit in the category of hausdorff lc vector space will be the quotient of $F$ by the closure of of $H$: a cone for the inductive limit is the same as a continuous map on $F$ which vanish on $H$, so if the target is hausdorff as well then it vanish on the closure.

The question of whether $H$ is closed or not is only relevant to know if the forgetful functor to the category of vector spaces or of general locally convex vector space commute to this specific inductive limit.