Added universality of the zero-dimensional spaces.

This commit is contained in:
Bokuan Li
2026-08-06 15:59:15 -04:00
parent a493db41c9
commit 3ba29569ef
7 changed files with 180 additions and 1 deletions

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@@ -135,7 +135,7 @@
\label{definition:seminorm-topology}
Let $E$ be a vector space over $K \in \RC$ and $\seqi{[\cdot]}$ be seminorms, then:
\begin{enumerate}
\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 For each $i \in I$, $d_i: E \times E \to [0, \infty)$ defined by $(x, y) \mapsto [x - y]_i$ is a pseudometric.
\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}