Banach–Alaoglu Theorem for a Separable Normed Space: the Closed Balls of the Dual Are Weak-Star Sequentially Compact and Compact for a Metric of Weak-Star Convergence

In the setting of Weak-Star Compactness and Vague Convergence: Standing Notation, let $X$ be a separable real normed space and let $M\in\mathbb{R}$ with $M\ge0$.

(i) Sequential compactness. Every sequence $(F_{m})_{m\in\mathbb{N}}$ in $B^{*}_{M}$ has a subsequence $(F_{m_{j}})_{j\in\mathbb{N}}$ that converges weak-star to some $F\in B^{*}_{M}$.

Let $(x_{k})_{k\in\mathbb{N}}$ be a sequence in $X$ with $\|x_{k}\|\le1$ for every $k\in\mathbb{N}$ whose range is dense in the closed unit ball $\{x\in X:\|x\|\le1\}$: for every $x\in X$ with $\|x\|\le1$ and every $\delta>0$ there is $k\in\mathbb{N}$ with $\|x-x_{k}\|<\delta$. Let

$$d(F,G)=\sup_{k\in\mathbb{N}}2^{-k}\,|F(x_{k})-G(x_{k})|\qquad(F,G\in B^{*}_{M}).$$

(ii) A metric for weak-star convergence. The suprema defining $d$ exist, $d$ is a metric on $B^{*}_{M}$, and a sequence $(G_{m})_{m\in\mathbb{N}}$ in $B^{*}_{M}$ converges to $G\in B^{*}_{M}$ in the metric space $(B^{*}_{M},d)$ if and only if $(G_{m})$ converges weak-star to $G$.

(iii) Compactness. $B^{*}_{M}$ is compact in the topological space $(B^{*}_{M},\mathcal{T}_{d})$.