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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.

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.

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 $$\varphi(g)=\lim_{k\to\infty} \varphi|_{X_n}(f_k)=\lim_{k\to\infty} \varphi|_{X_1}(f_k)=0.$$

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.

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.

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 $$\varphi(g)=\lim_{k\to\infty} \varphi|_{X_n}(f_k)=\lim_{k\to\infty} \varphi|_{X_1}(f_k)=0.$$

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.

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.

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 $$\varphi(g)=\lim_{k\to\infty} \varphi|_{X_n}(f_k)=\lim_{k\to\infty} \varphi|_{X_1}(f_k)=0.$$

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.

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.

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 $$\varphi(g)=\lim_{k\to\infty} \varphi|_{X_n}(f_k)=\lim_{k\to\infty} \varphi|_{X_1}(f_k)=0.$$

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

Edit. As spotted by Pietro Majer, the arguments below do not work because the functions $g_{z_k}$ need not belong to a fixed $H(U_n)$ if $z\in\bigcap_{n\in\mathbb N} \overline U_n$.

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.

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

Let $U_n$ be a decreasing sequence of non-empty, bounded, connected, open sets in $\mathbb C$ with empty intersection 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. To see this we define, for each $z\in\mathbb C$, the function $g_z(\omega)= 1/({z-\omega})$ which belongs to $X_n=H(U_n)$ if $z\notin U_n$. Then $h(z)=\varphi(g_z)$ defines a function on $\mathbb C$ which is holomorphic (since $(g_{z_k} - g_z)/(z_k-z)$ converges in $X_n$ if $z_k\to z$ in $U_n^c$) with $h(z_k)\to 0$ for $|z_k|\to\infty$ (since $g_{z_k}\to 0$ even in $X_1$). By Liouville's theorem, $h$ vanishes identically. Moreover, by Runge's theorem, $\{g_z: z\notin U_n\}$ has dense linear span in $X_n$ which implies that $\varphi$ vanishes on $X_n$.

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.

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.

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 $$\varphi(g)=\lim_{k\to\infty} \varphi|_{X_n}(f_k)=\lim_{k\to\infty} \varphi|_{X_1}(f_k)=0.$$