\section{Separable Normed Vector Spaces}
\label{section:separable-banach-space}

\begin{proposition}
\label{proposition:separable-dual}
    Let $K \in \RC$ and $\dpn{E, F}{\lambda}$ be a duality of normed vector spaces over $K$ with $E$ being separable, then
    \begin{enumerate}
        \item The closed unit ball $S = \bracsn{y \in F|\ \norm{y}_{F} \le 1}$ is separable with respect to the $\sigma(F, E)$-topology. 
        \item If the duality is norming, then there exists $\seq{y_n} \subset F$ such that for each $x \in E$, $\norm{x}_E = \sup_{n \in \natp}|\dpn{x, y_n}{\lambda}|$. 
    \end{enumerate}
\end{proposition}
\begin{proof}
    Let $\seq{x_n} \subset E$ be a dense subset. For each $N \in \natp$, let
    \[
    T_N: S \to \real^N \quad y \mapsto (\dpn{x_1, y}{E}, \cdots, \dpn{x_N, y}{E})
    \]

    Since $\real^N$ is separable, $T_N(S)$ is separable by \autoref{proposition:separable-metric-space}. Thus there exists $\bracs{y_{N, k}}_{k = 1}^\infty \subset S$ such that $\bracs{T_Ny_{N, k}}_{k = 1}^\infty$ is dense in $T_N(S)$.

    Let $y \in S$, then for each $N \in \natp$, there exists $k_N \in \natp$ such that for each $1 \le n \le N$, 
    \[
    |\dpn{x_n, \phi_{N, k_N}}{E} - \dpn{x_n, \phi}{E}| \le \frac{1}{N}
    \] 

    Thus for each $N \in \natp$, $\dpn{x_n, y_{N, k_N}}{E} \to \dpn{x_n, y}{E}$ as $N \to \infty$. Since $\phi_{N, k_N} \to \phi$ pointwise on a dense subset of $E$, and $\bracsn{y_{N, k_N}|N \in \natp} \subset S$ is uniformly equicontinuous, $\phi_{N, k_N} \to \phi$ in the $\sigma(F, E)$-topology by \autoref{proposition:strong-operator-dense}. 
\end{proof}

\begin{proposition}
\label{proposition:separable-banach-borel-sigma-algebra}
    Let $E$ be a separable normed vector space, then the Borel $\sigma$-algebra on $E$ is generated by the following families of sets:
    \begin{enumerate}
        \item Open sets in $E$ with respect to the strong topology. 
        \item $\bracs{B(x, r)|x \in E, r > 0}$.
        \item $\bracsn{\ol{B(x, r)}|x \in E, r > 0}$. 
        \item Open sets in $E$ with respect to the weak topology. 
    \end{enumerate}

    
\end{proposition}
\begin{proof}
    (1) $\Leftrightarrow$ (2) $\Leftrightarrow$ (3): By \autoref{proposition:separable-metric-borel-sigma-algebra}.

    (4) $\subset$ (1): Every weakly open set is strongly open. 

    (2) $\subset$ (4): By \autoref{proposition:seminorm-lsc}, $\norm{\cdot}_E: E \to [0, \infty)$ is Borel measurable with respect to the weak topology. For any $x \in E$, let
    \[
    \phi_x: E \to [0, \infty) \quad y \mapsto \norm{x - y}_E
    \]

    then $\phi_x$ is Borel measurable with respect to the weak topology, so $B(x, r) = \bracs{\phi_x < r}$ is a Borel set with respect to the weak topology. 
\end{proof}