Fixed regex incident.
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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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@@ -138,7 +138,7 @@
\item For each $i \in I$, $d_i: E \times E \to [0, \infty)$ defined by $(x, y) \mapsto [x - y]_i$ is a pseudo-metric.
\item The topology induced by $\seqi{d}$ makes $E$ a topological vector space.
\item For each $i \in I$, $[\cdot]_i: E \to [0, \infty)$ is continuous.
\end\{enumerate\}
\end{enumerate}
The topology induced by $\seqi{d}$ is the \textbf{vector space topology induced by} $\seqi{[\cdot]}$. In addition,
\begin{enumerate}
@@ -160,7 +160,7 @@
\item If $A$ is convex, then for any $x, y \in E$, $[x + y]_A \le [x]_A + [y]_A$.
\item If $A$ is circled, then for any $x \in E$ and $\lambda \in K$, $[\lambda x]_A = \abs{\lambda}[x]_A$.
\item If $A$ is circled, then $\bracs{\rho < 1} \subseteq A \subseteq \bracs{\rho \le 1} \subseteq \ol A$.
\end\{enumerate\}
\end{enumerate}
In particular,
\begin{enumerate}[start=4]
@@ -191,7 +191,7 @@
\item There exists a fundamental system of neighborhoods at $0$ consisting of convex sets.
\item There exists a fundamental system of neighbourhoods at $0$ consisting of convex, circled, and radial sets.
\item There exists a family of seminorms $\seqi{[\cdot]}$ that induces the topology on $E$.
\end\{enumerate\}
\end{enumerate}
If the above holds, then $E$ is a \textbf{locally convex} space.
\end{definition}