\section{Vector Space Topologies}
\label{section:tvs-topology}


\begin{definition}[Topological Vector Space]
\label{definition:tvs}
    Let $E$ be a vector space over $K \in \bracs{\real, \complex}$ and $\topo \subset 2^E$ be a topology. If
    \begin{enumerate}
        \item[(TVS1)] The addition map $E \times E \to E$ with $(x, y) \mapsto x + y$ is continuous.
        \item[(TVS2)] The scalar multiplication map $K \times E \to E$ with $(\lambda, x) \mapsto \lambda x$ is continuous.
    \end{enumerate}

    then the pair $(E, \topo)$ is a \textbf{topological vector space}.
\end{definition}

\begin{lemma}
\label{lemma:tvs-translation-invariant}
    Let $E$ be a TVS over $K \in \RC$, then the topology of $E$ is translation-invariant.
\end{lemma}
\begin{proof}
    Let $U \subset E$ open and $y \in E$, then $U + y$ is the preimage of $U$ by the map $x \mapsto x - y$. By (TVS1), $U + y$ is open.
\end{proof}


\begin{proposition}
\label{proposition:tvs-uniform}
    Let $E$ be a TVS over $K \in \bracs{\real, \complex}$, then:
    \begin{enumerate}
        \item There exists a unique translation-invariant uniformity $\fU$ on $E$ that induces the topology on $E$.
        \item For each neighbourhood $V \in \cn(0)$, let $U_V = \bracs{(x, y) \in E^2| x - y \in V}$, then for any fundamental system of neighbourhoods $\fB_0$ at $0$, $\fB = \bracs{U_V| V \in \fB_0}$ is a fundamental system of entourages for $\fU$.
    \end{enumerate}

    The space $E$ will always be assumed to be equipped with its translation-invariant uniformity.
\end{proposition}
\begin{proof}
    By \autoref{definition:group-translation-invariant-uniformity}. 
\end{proof}

\begin{definition}[Balanced/Circled]
\label{definition:balanced}
    Let $E$ be a vector space over $K \in \RC$ and $A \subset E$, then $A$ is \textbf{balanced/circled} if $\lambda A \subset A$ for all $\lambda \in K$ with $\abs{\lambda} \le 1$.
\end{definition}

\begin{definition}[Absorbing/Radial]
\label{definition:absorbing}
    Let $E$ be a vector space over $K \in \RC$ and $A, B \subset E$, then $A$ \textbf{absorbs} $B$ if there exists $\lambda \in K$ such that $\lambda A \supset B$, and $A$ is \textbf{absorbing/radial} if it absorbs every point in $E$.
\end{definition}

\begin{proposition}
\label{proposition:tvs-good-neighbourhood-base}
    Let $E$ be a topological vector space over $K \in \RC$, then
    \begin{enumerate}
        \item $E$ admits a fundamental system of neighbourhoods at $0$ consisting of circled and radial sets.
        \item The fundamental system of neighbourhoods in $(1)$ can be taken to be open or closed.
    \end{enumerate}
\end{proposition}
\begin{proof}
    Firstly, (TVS2) implies that every neighbourhood of $0$ is circled.

    By \autoref{proposition:uniform-neighbourhoods}, $E$ admits a fundamental system of neighbourhoods consisting of open sets or closed sets.

    Let $U \in \cn^o(0)$ be open. By (TVS2), there exists $r > 0$ such that $\lambda U \subset U$ for all $\lambda \in K$ with $\abs{\lambda} \le r$. Define
    \[
    V = \bigcup_{\substack{\lambda \in K \\ \abs{\lambda} \le r}} \lambda U \subset U
    \]

    then for any $x \in V$, there exists $\lambda \in K$ with $\abs{\lambda} \le r$ and $y \in U$ such that $x = \lambda y$. In which case, for any $\mu \in K$ with $\abs{\mu} \le 1$, $\mu(\lambda y) = (\mu \lambda) y \in \mu\lambda U$. Since $\abs{\mu \lambda} \le r$, $\mu \lambda U \subset V$. Thus $V \subset U$ is balanced.

    Let $U \in \cn(0)$ be closed, then there exists a balanced neighbourhood $V \in \cn^o(0)$ such that $V \subset U$. In which case, for any $\lambda \in K$ with $0 < \abs{\lambda} \le 1$, $\lambda \overline{V} = \overline{\lambda V} \subset \overline{V}$ by (TVS2). Therefore $\overline{V} \subset U$ is balanced as well.
\end{proof}

\begin{proposition}
\label{proposition:tvs-0-neighbourhood-base}
    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 $E$ 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}[Proof, {{\cite[I.1.2]{SchaeferWolff}}}. ]
    \textbf{Forward:} By \autoref{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{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
    \[
    \mathfrak{V} = \bracs{U_V|V \in \fB}
    \]

    then
    \begin{enumerate}
        \item[(FB1)] For any $V, V' \in \fB$, there exists $W \in \fB$ with $W \subset V \cap V'$. In which case, $U_{V} \cap U_{V'} \supset U_W \in \mathfrak{V}$.
        \item[(UB1)] For any $x \in E$ and $V \in \fB$, $x - x = 0 \in V$, so $\Delta \subset U_V$.
        \item[(UB3)] For any $V \in \fB$, by (TVB1), there exists $W \in \fB$ such that $W + W \subset V$. In which case, for any $x, y, z \in E$ with $x - y, y - z \in W$, $x - z \in V$. Therefore $U_W \circ U_W \subset U_V$.
    \end{enumerate}

    By \autoref{proposition:fundamental-entourage-criterion}, there exists a unique uniformity $\fU$ on $E$ for which $\mathfrak{V}$ is a fundamental system of entourages.

    (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
        \begin{align*}
            \lambda x - \lambda' x' &= \lambda x - \lambda x' + \lambda x' - \lambda' x' \\
            &= \lambda(x - x') + (\lambda - \lambda')x' \in \lambda V + (\lambda - \lambda')x'
        \end{align*}
        Since $V$ is radial, there exists $\mu \in K$ such that $x \in \mu V$. Given that $V$ is circled, $x' = x + (x - x') \in \mu V + V \subset (\abs{\mu} + 1)V$, and
        \[
        \lambda x - \lambda' x' \in \lambda V + \eps(\abs \mu + 1)V \subset \abs{\lambda} V + \eps(\abs \mu + 1)V
        \]


        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}

    (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 \autoref{proposition:neighbourhoodcharacteristic}, $\mathcal{S} = \mathcal{T}$.
\end{proof}


\begin{proposition}
\label{proposition:tvs-locally-connected}
    Let $E$ be a TVS over $K \in \RC$, then $E$ is locally connected.
\end{proposition}
\begin{proof}
    Let $U \in \cn(0)$ be radial, then for any $y \in U$, the mapping $t \mapsto ty$ is a path from $0$ to $y$ contained in $U$. Thus $U$ is path-connected. By \autoref{proposition:tvs-0-neighbourhood-base}, the radial neighbourhoods of $0$ forms a fundamental system of neighbourhoods, so the path-connected neighbourhoods of $0$ forms a fundamental system as well.
\end{proof}