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Added quotient spaces.

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Bokuan Li
2026-01-23 16:44:29 -05:00
parent f4ffd7d838
commit 220c263f8f
3 changed files with 68 additions and 6 deletions
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src/fa/tvs/definition.tex
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@@ -150,17 +150,25 @@
\begin{proposition}[{{\cite[1.2]{SchaeferWolff}}}]
\label{proposition:tvs-0-neighbourhood-base}
Let $E$ be a vector space of $K \in \RC$, and $\topo$ be a vector space topology on $E$, then there exists a fundamental system of neighbourhoods $\fB \subset \cn_E(0)$ such that:
Let $E$ be a vector space over $K \in \RC$, and $\topo$ be a vector space topology on $E$, then there exists a fundamental system of neighbourhoods $\fB \subset \cn_E(0)$ such that:
\begin{enumerate}
\item[(TVB1)] For each $U \in \fB$, there exists $V \in \fB$ such that $V + V \subset U$.
\item[(TVB2)] For each $U \in \fB$, $U$ is circled and radial.
\end{enumerate}
Conversely, if $\fB \subset 2^E$ is a family of sets that contain $0$ and satisfies (TVB1) and (TVB2), then there exists a unique topology $\topo$ on $E$ for which $\fB$ is a fundamental system of neighbourhoods at $0$.
Conversely, if $\fB \subset 2^E$ is a family of sets that contain $0$ and satisfies (TVB1) and (TVB2), then there exists a unique topology $\topo$ on $E$ such that:
\begin{enumerate}
\item $\topo$ is translation-invariant.
\item $\fB$ is a fundamental system of neighbourhoods at $0$ for $\topo$.
\end{enumerate}
Moreover,
\begin{enumerate}
\item[(3)] $(E, \topo)$ is a TVS.
\end{enumerate}
\end{proposition}
\begin{proof}
(1): By \ref{proposition:tvs-good-neighbourhood-base}, there exists a fundamental system of neighbourhoods $\fB \subset \cn_E(0)$ consisting of circled and radial sets. By (TVS1), $\fB$ satisfies (TVB1).
\textbf{Forward:} By \ref{proposition:tvs-good-neighbourhood-base}, there exists a fundamental system of neighbourhoods $\fB \subset \cn_E(0)$ consisting of circled and radial sets. By (TVS1), $\fB$ satisfies (TVB1).
(2): For each $V \in \fB$, let $U_V = \bracs{(x, y) \in E|x - y \in V}$, then $U_V$ is symmetric and translation-invariant by (TVB1). Let
\textbf{Converse:} For each $V \in \fB$, let $U_V = \bracs{(x, y) \in E|x - y \in V}$, then $U_V$ is symmetric and translation-invariant by (TVB1). Let
\[
\mathfrak{V} = \bracs{U_V|V \in \fB}
\]
@@ -172,7 +180,11 @@
\end{enumerate}
By \ref{proposition:fundamental-entourage-criterion}, there exists a unique uniformity $\fU$ on $E$ for which $\mathfrak{V}$ is a fundamental system of entourages.
It remains to verify that $\mathfrak{V}$ induces a vector space topology on $E$.
(1): Since $\mathfrak{V}$ is translation-invariant, so is $\fU$.
(2): By definition of the uniform topology, $\fB = \bracs{U_V(0)|V \in \fB}$ is a fundamental system of neighbourhoods at $0$.
(3):
\begin{enumerate}
\item[(TVS1)] Let $V \in \fB$, then there exists $W \in \fB$ such that $W + W \subset V$ by (TVB1). In which case, for any $x, x', y, y'$ with $x - x' \in W$ and $y - y' \in W$, $(x + y) - (x' + y') \in W + W \subset V$.
\item[(TVS2)] Let $V \in \fB$, $\eps > 0$, $x, x' \in E$ with $x - x' \in V$, and $\lambda, \lambda' \in K$ with $\abs{\lambda - \lambda'} < \eps$, then
@@ -188,5 +200,5 @@
Let $W \in \fB$ and $\eps > 0$ such that $\eps(\abs{\mu}+1) \le 1$, then by repeated application of (TVB1), there exists $V \in \fB$ such that $\abs{\lambda} V + V \subset W$. Therefore scalar multiplication is jointly continuous.
\end{enumerate}
Finally, by definition of the uniform topology, $\fB = \bracs{U_V(0)|V \in \fB}$ is a fundamental system of neighbourhoods at $0$.
(Uniqueness): Let $\mathcal{S} \subset 2^E$ be a topology on $E$ satisfying (1) and (2), then for each $x \in E$, $\cn_{(E, \mathcal{S})}(x) = \cn_{(E, \mathcal{T})}(x)$. By \ref{proposition:neighbourhoodcharacteristic}, $\mathcal{S} = \mathcal{T}$.
\end{proof}