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.



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


There are exmples 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 $X\_1$ 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
$$\varphi(g)=\lim\_{k\to\infty} \varphi|\_{X\_n}(f\_k)=\lim\_{k\to\infty} \varphi|\_{X\_1}(f\_k)=0.$$


