10.4 Separable Normed Vector Spaces
Proposition 10.4.1. Let $E$ be a separable normed vector space, then $E^{*}$ is separable with respect to the weak*-topology.
Proof. Let $\seq{x_n}\subset E$ be a dense subset and $S = \bracsn{\phi \in E^*| \norm{\phi}_{E^*} \le 1}$. For each $N \in \natp$, let
Since $\real^{N}$ is separable, $T_{N}(S)$ is separable by Proposition 7.1.2. Thus there exists $\bracs{\phi_{N, k}}_{k = 1}^{\infty} \subset S$ such that $\bracs{T_N\phi_{N, k}}_{k = 1}^{\infty}$ is dense in $T_{N}(S)$.
Let $\phi \in S$, then for each $N \in \natp$, there exists $k_{N} \in \natp$ such that for each $1 \le n \le N$,
Thus for each $N \in \natp$, $\dpn{x_n, \phi_{N, k_N}}{E}\to \dpn{x_n, \phi}{E}$ as $N \to \infty$. Since $\phi_{N, k_N}\to \phi$ pointwise on a dense subset of $E$, and $\bracsn{\phi_{N, k_N}|N \in \natp}\subset S$ is uniformly equicontinuous, $\phi_{N, k_N}\to \phi$ in the weak*-topology by Proposition 8.11.7.$\square$
Proposition 10.4.2. Let $E$ be a separable normed vector space, then the Borel $\sigma$-algebra on $E$ is generated by the following families of sets:
Open sets in $E$ with respect to the strong topology.
$\bracs{B(x, r)|x \in E, r > 0}$.
$\bracsn{\ol{B(x, r)}|x \in E, r > 0}$.
Open sets in $E$ with respect to the weak topology.
Proof. (1) $\Leftrightarrow$ (2) $\Leftrightarrow$ (3): By Proposition 14.2.5.
(4) $\subset$ (1): Every weakly open set is strongly open.
(2) $\subset$ (4): By Proposition 9.7.7, $\norm{\cdot}_{E}: E \to [0, \infty)$ is Borel measurable with respect to the weak topology. For any $x \in E$, let
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.$\square$