Added quotient spaces.

src/fa/tvs/definition.tex

@@ -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}

src/fa/tvs/index.tex

@@ -5,5 +5,6 @@
 \input{./src/fa/tvs/bounded.tex}
 \input{./src/fa/tvs/dual.tex}
 \input{./src/fa/tvs/continuous.tex}
+\input{./src/fa/tvs/quotient.tex}
 \input{./src/fa/tvs/completion.tex}
 \input{./src/fa/tvs/spaces-of-linear.tex}

src/fa/tvs/quotient.tex

@@ -0,0 +1,49 @@
+\section{Quotient Spaces}
+\label{section:tvs-quotient}
+
+\begin{definition}[Quotient TVS]
+\label{definition:tvs-quotient}
+    Let $E$ be a TVS over $K \in \RC$, and $M \subset E$ be a vector subspace, then there exists $(\td E, \pi)$ such that:
+    \begin{enumerate}
+        \item $\td E$ is a TVS over $K \in \RC$.
+        \item $\pi \in L(E; \td E)$.
+        \item $\ker \pi \supset M$.
+        \item[(U)] For any topological space $F$ and $f \in C(E;F)$ such that $f(x) = f(y)$ whenever $x - y \in M$, there exists a unique $\td f \in C(\td E; F)$ such that the following diagram commutes
+        \[
+        \xymatrix{
+        E \ar@{->}[rd]^{f} \ar@{->}[d]_{\pi} &  \\
+        \widetilde E \ar@{->}[r]_{\tilde f} & F
+        }
+        \]
+        If $F$ is a TVS over $K$ and $f \in L(E; F)$, then $\td f \in L(E; F)$.
+    \end{enumerate}
+    The space $\td E = E/M$ is the \textbf{quotient} of $E$ by $M$.
+\end{definition}
+\begin{proof}
+    Let $\td E = E/M$ be the algebraic quotient of $E$ by $M$, and equip it with the quotient topology by $\pi$.
+
+    (1): By \ref{definition:quotient-topology}, for each $\pi(U) \subset E/M$, $\pi(U)$ is open if and only if $U$ is open. Since the topology on $E$ is translation-invariant, so is the quotient topology on $E/M$. Let
+    \[
+    \fB = \bracs{\pi(U)| U \in \cn(0) \text{ circled and radial}}
+    \]
+    Since the circled and radial neighbourhoods of $0$ forms a fundamental system of neighbourhoods at $0$ by \ref{proposition:tvs-good-neighbourhood-base}, $\fB$ is a fundamental system of neighbourhoods at $0$ for the quotient topology on $E/M$. In addition,
+    \begin{enumerate}
+        \item[(TVB1)] Let $U \in \cn(0)$ be circled and radial. For any $\lambda \in K$ with $\abs{\lambda} \le 1$, $\lambda \pi(U) = \pi(\lambda U) \subset \pi(U)$, so $\pi(U)$ is also circled. For any $x + M \in E/M$, there exists $\lambda \in K$ such that $x \in \lambda U$. In which case, $x \in \lambda U + M = \pi(U)$, so $\pi(U)$ is also radial.
+        \item[(TVB2)] For any $U \in \cn(0)$ circled and radial, by \ref{proposition:tvs-good-neighbourhood-base}, there exists $W \in \cn(0)$ such that $W + W \subset U$. In which case, $\pi(W) + \pi(W) \subset \pi(U)$.
+    \end{enumerate}
+    By \ref{proposition:tvs-0-neighbourhood-base}, there exists a unique translation-invariant topology on $E/M$ such that $\fB$ is a fundamental system of neighbourhoods at $0$, which must be the quotient topology on $E/M$. In which case, the quotient topology is a vector space topology by (3) of \ref{proposition:tvs-0-neighbourhood-base}.
+
+    (2), (3), (U): By \ref{definition:quotient-topology}.
+\end{proof}
+
+\begin{proposition}
+\label{proposition:tvs-quotient-hausdorff}
+    Let $E$ be a TVS over $K \in \RC$, $M \subset E$ be a subspace, then $E/M$ is Hausdorff if and only if $M$ is closed.
+\end{proposition}
+\begin{proof}
+    The space $M$ is closed if and only if
+    \[
+    M = \bigcap_{V \in \cn(0)}M + V
+    \]
+    which is equivalent to $E/M$ being Hausdorff.
+\end{proof}