[proofplan]
We prove the equivalence of three characterisations of distributions: topological continuity (1), sequential continuity (2), and the seminorm bound (3).
The implication (1) $\Rightarrow$ (2) is a general topological fact.
The implication (2) $\Rightarrow$ (3) uses two key ingredients: first, sequential continuity on a Fréchet space implies continuity (because Fréchet spaces are metrizable); second, a continuous linear functional on a locally convex space admits a seminorm bound.
The implication (3) $\Rightarrow$ (1) follows from the universal property of the inductive limit topology.
[/proofplan]
[step:Show (1) $\Rightarrow$ (2): topological continuity implies sequential continuity]
Let $\varphi_k \to \varphi$ in $\mathcal{D}(\Omega)$.
For any open $U \subseteq \mathbb{R}$ containing $T(\varphi)$, continuity of $T$ gives $T^{-1}(U)$ is open in $\mathcal{D}(\Omega)$.
Since $\varphi \in T^{-1}(U)$ and $\varphi_k \to \varphi$, eventually $\varphi_k \in T^{-1}(U)$, so $T(\varphi_k) \in U$.
Since $U$ was arbitrary: $T(\varphi_k) \to T(\varphi)$.
[/step]
[step:Show (2) $\Rightarrow$ (3): sequential continuity yields the seminorm bound]
Fix a compact set $K \subset \Omega$.
The restriction $T|_{\mathcal{D}_K(\Omega)}$ is a linear functional on the Fréchet space $\mathcal{D}_K(\Omega)$, equipped with the seminorms $p_{K,N}(\varphi) := \sum_{|\alpha| \le N} \sup_K |\partial^\alpha \varphi|$.
[claim:Sequential continuity implies continuity on the Fréchet step]
The restriction $T|_{\mathcal{D}_K(\Omega)}$ is continuous.
[/claim]
[proof]
Sequential convergence in $\mathcal{D}_K(\Omega)$ implies sequential convergence in $\mathcal{D}(\Omega)$ (since the inclusion $\iota_K: \mathcal{D}_K(\Omega) \hookrightarrow \mathcal{D}(\Omega)$ is continuous).
By hypothesis (2), $T$ is sequentially continuous on $\mathcal{D}(\Omega)$, so $T|_{\mathcal{D}_K(\Omega)} = T \circ \iota_K$ is sequentially continuous.
The Fréchet space $\mathcal{D}_K(\Omega)$ is metrizable: the topology is generated by the countable family $\{p_{K,N}\}_{N \in \mathbb{N}_0}$, which 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*}
In a metrizable space, sequential continuity implies continuity (since a set is closed iff it is sequentially closed, and preimages of closed sets under a sequentially continuous map are sequentially closed, hence closed).
Therefore $T|_{\mathcal{D}_K(\Omega)}$ is continuous.
[/proof]
[claim:A continuous linear functional admits a seminorm bound]
There exist $N_K \in \mathbb{N}_0$ and $C_K > 0$ such that $|T(\varphi)| \leq C_K \, p_{K,N_K}(\varphi)$ for all $\varphi \in \mathcal{D}_K(\Omega)$.
[/claim]
[proof]
Continuity of $T|_{\mathcal{D}_K(\Omega)}$ at the origin means $T^{-1}((-1,1))$ is open in $\mathcal{D}_K(\Omega)$.
By the definition of the locally convex topology, every open neighbourhood of $0$ contains a set of the form $\{\varphi : p_{K,N}(\varphi) < \varepsilon\}$ for some $N$ and $\varepsilon > 0$.
So there exist $N_K \in \mathbb{N}_0$ and $\varepsilon > 0$ with
\begin{align*}
p_{K,N_K}(\varphi) < \varepsilon \implies |T(\varphi)| < 1.
\end{align*}
For $\varphi \in \mathcal{D}_K(\Omega)$ 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 bound.
If $p_{K,N_K}(\varphi) = 0$, then for any $\delta > 0$ and $\psi_0$ 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 small $\delta$, $|T(\varphi + \delta\psi_0)| < 1$.
Sending $\delta \to 0$: $|T(\varphi)| \leq 1$, and the same rescaling gives $|T(\varphi)| = 0 \leq C_K\, p_{K,N_K}(\varphi)$.
[/proof]
Combining the two claims and expanding $p_{K,N_K}$: for every compact $K \subset \Omega$, there exist $N_K$ and $C_K$ such that
\begin{align*}
|T(\varphi)| \leq C_K \sum_{|\alpha| \leq N_K} \sup_{x \in K} |\partial^\alpha \varphi(x)| \quad \text{for all } \varphi \in \mathcal{D}_K(\Omega),
\end{align*}
which is condition (3).
[/step]
[step:Show (3) $\Rightarrow$ (1): the seminorm bound implies topological continuity via the universal property]
By the universal property of the [strict inductive limit topology](/page/Strict%20Inductive%20Limit%20Topology), $T: \mathcal{D}(\Omega) \to \mathbb{R}$ is continuous if and only if $T \circ \iota_K: \mathcal{D}_K(\Omega) \to \mathbb{R}$ is continuous for every compact $K \subset \Omega$.
Fix a compact $K$.
Condition (3) provides $N_K$ and $C_K$ with $|T(\varphi)| \leq C_K\, p_{K,N_K}(\varphi)$ 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 $\mathcal{D}_K(\Omega)$.
A linear functional dominated by a continuous seminorm is continuous: for any open $U \subseteq \mathbb{R}$ containing $0$, choose $\delta > 0$ with $(-\delta, \delta) \subseteq U$.
Then $\{\varphi : p_{K,N_K}(\varphi) < \delta/C_K\}$ is open in $\mathcal{D}_K(\Omega)$ and maps into $U$.
Since this holds for every $K$, the universal property gives that $T$ is continuous on $\mathcal{D}(\Omega)$.
[/step]
