**(1) $\Rightarrow$ (2).** This is a general topological fact. Let $\varphi_k \to \varphi$ in $\mathcal{D}(\Omega)$. The preimage $T^{-1}(U)$ of every open set $U \subseteq \mathbb{R}$ is open in $\mathcal{D}(\Omega)$ by continuity. Since $\varphi_k \to \varphi$ and $\varphi \in T^{-1}(U)$ for any open $U$ containing $T(\varphi)$, eventually $\varphi_k \in T^{-1}(U)$, i.e. $T(\varphi_k) \in U$. Since $U$ was an arbitrary open neighbourhood of $T(\varphi)$, this gives $T(\varphi_k) \to T(\varphi)$.

**(2) $\Rightarrow$ (3).** Fix a compact set $K \subset \Omega$. The restriction $T|_{\mathcal{D}_K(\Omega)}$ is a linear functional on the [Fréchet space $\mathcal{D}_K(\Omega)$](/page/Test%20Function), which is equipped with the seminorms $p_{K,N}(\varphi) := \sum_{|\alpha| \le N} \sup_K |\partial^\alpha \varphi|$ for $N \in \mathbb{N}_0$.

[claim: Sequential Continuity Implies Continuity On Fréchet Spaces] Let $T|_{\mathcal{D}_K(\Omega)}: \mathcal{D}_K(\Omega) \to \mathbb{R}$ be a sequentially continuous linear functional on the Fréchet space $\mathcal{D}_K(\Omega)$. Then $T|_{\mathcal{D}_K(\Omega)}$ is continuous. [/claim]

[proof] Since $\mathcal{D}_K(\Omega)$ is a Fréchet space, its topology is generated by the countable family of seminorms $\{p_{K,N}\}_{N \in \mathbb{N}_0}$. A countable family of seminorms induces a translation-invariant metric \begin{align*} d(\varphi, \psi) &:= \sum_{N=0}^{\infty} 2^{-N} \frac{p_{K,N}(\varphi - \psi)}{1 + p_{K,N}(\varphi - \psi)}, \end{align*} which generates the same topology. In any metrizable topological space, continuity and sequential continuity of a map are equivalent: a set is closed if and only if it is sequentially closed (since metrizability implies first-countability), so the preimage of a closed set under a sequentially continuous map is sequentially closed, hence closed. Applying this to $T|_{\mathcal{D}_K(\Omega)}$, which is sequentially continuous by hypothesis (2) (sequential convergence in $\mathcal{D}_K(\Omega)$ implies sequential convergence in $\mathcal{D}(\Omega)$ by continuity of the inclusion $\iota_K$), gives that $T|_{\mathcal{D}_K(\Omega)}$ is continuous. [/proof]

[claim: Continuous Linear Functional Admits Seminorm Bound] If $T|_{\mathcal{D}_K(\Omega)}$ is continuous, then there exist $N_K \in \mathbb{N}_0$ and $C_K > 0$ such that \begin{align*} |T(\varphi)| &\leq C_K \, p_{K,N_K}(\varphi) \quad \text{for all } \varphi \in \mathcal{D}_K(\Omega). \end{align*} [/claim]

[proof] Continuity of $T|_{\mathcal{D}_K(\Omega)}$ at the origin means that $T^{-1}((-1,1))$ is an open neighbourhood of $0$ in $\mathcal{D}_K(\Omega)$. By the definition of the locally convex topology generated by $\{p_{K,N}\}$, every open neighbourhood of $0$ contains a basic neighbourhood of the form $\{\varphi : p_{K,N}(\varphi) < \varepsilon\}$ for some $N \in \mathbb{N}_0$ and $\varepsilon > 0$. So there exist $N_K \in \mathbb{N}_0$ and $\varepsilon > 0$ such that \begin{align*} p_{K,N_K}(\varphi) < \varepsilon \implies |T(\varphi)| < 1. \end{align*} Now let $\varphi \in \mathcal{D}_K(\Omega)$ be arbitrary with $p_{K,N_K}(\varphi) \neq 0$. Set $\psi := \varepsilon \varphi / (2 \, p_{K,N_K}(\varphi))$. Then $p_{K,N_K}(\psi) = \varepsilon/2 < \varepsilon$, so $|T(\psi)| < 1$. By linearity, \begin{align*} |T(\varphi)| &= \frac{2 \, p_{K,N_K}(\varphi)}{\varepsilon} \, |T(\psi)| < \frac{2}{\varepsilon} \, p_{K,N_K}(\varphi). \end{align*} Setting $C_K := 2/\varepsilon$ gives the result. The case $p_{K,N_K}(\varphi) = 0$ follows by a limiting argument: for any $\delta > 0$ and any $\psi_0$ with $p_{K,N_K}(\psi_0) < \varepsilon$, the function $\varphi + \delta \psi_0$ satisfies $p_{K,N_K}(\varphi + \delta \psi_0) = \delta \, p_{K,N_K}(\psi_0) < \delta\varepsilon$, so for $\delta$ small enough $|T(\varphi + \delta\psi_0)| < 1$; sending $\delta \to 0$ gives $|T(\varphi)| \leq 1$, and then the same rescaling gives $|T(\varphi)| = 0 \leq C_K \, p_{K,N_K}(\varphi)$. [/proof]

Combining the two claims and expanding $p_{K,N_K}$ gives: for every compact $K \subset \Omega$, there exist $N_K \in \mathbb{N}_0$ and $C_K > 0$ such that $|T(\varphi)| \leq C_K \sum_{|\alpha| \leq N_K} \sup_K |\partial^\alpha \varphi|$ for all $\varphi \in \mathcal{D}_K(\Omega)$, which is condition (3).

**(3) $\Rightarrow$ (1).** By the [definition of the strict inductive limit topology](/page/Test%20Function), $T: \mathcal{D}(\Omega) \to \mathbb{R}$ is continuous if and only if $T \circ \iota_K: \mathcal{D}_K(\Omega) \to \mathbb{R}$ is continuous for every compact $K \subset \Omega$, where $\iota_K: \mathcal{D}_K(\Omega) \hookrightarrow \mathcal{D}(\Omega)$ is the inclusion. This is the universal property of the inductive limit topology: a linear map out of the inductive limit is continuous if and only if its restriction to each constituent space is continuous.

Fix a compact $K \subset \Omega$. Condition (3) provides $N_K$ and $C_K$ such that $|T(\varphi)| \leq C_K \, p_{K,N_K}(\varphi)$ for all $\varphi \in \mathcal{D}_K(\Omega)$. This means $T \circ \iota_K$ is dominated by the continuous seminorm $C_K \, p_{K,N_K}$ on the Fréchet space $\mathcal{D}_K(\Omega)$. A linear functional dominated by a continuous seminorm is continuous: for any open set $U \subseteq \mathbb{R}$ containing $0$ there exists $\delta > 0$ with $(-\delta, \delta) \subseteq U$, and then $\{\varphi : p_{K,N_K}(\varphi) < \delta/C_K\}$ is an open neighbourhood of $0$ in $\mathcal{D}_K(\Omega)$ that maps into $U$. Since this holds for every $K$, the universal property gives that $T$ is continuous on $\mathcal{D}(\Omega)$.