Fixed regex incident.
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This commit is contained in:
Bokuan Li
2026-05-05 02:00:05 -04:00
parent 0f2e69d1f9
commit 97372173e1
78 changed files with 172 additions and 172 deletions

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@@ -82,7 +82,7 @@
\begin{enumerate}
\item[(a)] For each $n \in \natp$, $U_n$ is circled, radial, and contains $0$.
\item[(b)] For each $n \in \natp$, $U_{n+1} + U_{n+1} \subset U_n$.
\end\{enumerate\}
\end{enumerate}
then there exists a pseudonorm $\rho: E \to [0, \infty)$ such that for each $n \in \natp$,
\[
@@ -111,7 +111,7 @@
so $\rho(\lambda x) \le \rho(x)$.
\item[(PN3)] Let $x, y \in X$ and $M, N \subset \natp$ finite such that $x \in U_M$ and $y \in U_N$. Assume without loss of generality that $\rho_M + \rho_N < 1$, then there exists a unique $P \subset \nat$ finite such that $\rho_P = \rho_M + \rho_N$. In which case, $U_P \supset U_M + U_N$ by assumption (b). Therefore $\rho(x + y) \le \rho(x) + \rho(y)$.
\end\{enumerate\}
\end{enumerate}
For any $x \in U_{n+1}$, $\rho(x) \le 2^{-n+1} < 2^n$, so $U_{n+1} \subset \rho^{-1}([0, 2^{-n}))$ by \autoref{proposition:dyadic-semigroup-order}. On the other hand, for any $x \in E$ with $\rho(x) < 2^{-n}$, $x \in U_{2^{-n}} = U_n$. This allows showing the remaining seminorm axioms by considering neighbourhoods of the form $\bracs{U_n|n \in \natp}$.
\begin{enumerate}