Definition 1: Let $X$ be a normed space. The operator \begin{align*} J_{X}\colon X&\to X^{**}\\ x&\mapsto J_{X}(x)(x^{*})=x^{*}(x) \end{align*} is called canonical lace (or canonical immersion) of $X$ into $X^{**}. $
Lemma 1: Let $X$ be a normed space. Then the canonical immersion $J_{X}$ of $X$ into $X^{∗∗}$ is an isometric isomorphism.
Definition 2: Let $ (X, \Vert \cdot \Vert) $ be a normed space. The weak- topology*, denoted by $ \sigma (X ^{*}, X), $ is the smallest topology at $ X ^{*} $ that makes every element of $ \{J_ {X} (x) \mid x \in X \} \subseteq {X ^{**}}$ continuous.
Let \begin{equation*} X=c_{00}=\{x=(x_{n})_{n\in\mathbb{N}}\in\mathbb{K}^{\mathbb{N}}\mid \exists N\in\mathbb{N}, x_{n}=0, \forall n\geq N\}. \end{equation*}
Question: Is the topological space $ (X^{*},\sigma (X ^ {*}, X)) $ metrizable?
Attempt: We know that $(c_{00},\Vert\cdot\Vert_{\infty})$ is a normed space, which is not Banach. A completion of $(c_{00},\Vert\cdot\Vert_{\infty})$ is \begin{equation*} c_{0}=\left\{y=(y_{n})_{n\in\mathbb{N}}\in\mathbb{K}^{\mathbb{N}}\ \middle|\ \lim_{n\rightarrow{\infty}}{x_{n}}=0\right\} \end{equation*} that is, there is $J\colon c_{00}\to c_{0}$ linear such that $\Vert J(x)\Vert=\Vert x\Vert,\forall x\in c_{00}$ and $\overline{J(c_{00})}=c_{0}.$
Furthermore, we know that \begin{equation*} X^{*}=c_{0}^{*}=l_{1}=\left\{z=(z_{n})_{n\in\mathbb{N}}\in\mathbb{K}^{\mathbb{N}}\ \middle|\ \sum_{n\in\mathbb{N}}{z_{n}}<\infty\right\}. \end{equation*} and \begin{equation*} l_{1}^{*}=l^{\infty}=\left\{w=(w_{n})\in\mathbb{K}^{\mathbb{N}} \ \middle|\ \sup_{n\in\mathbb{N}}\{|w_{n}|\}<\infty \right\}. \end{equation*}
I don't know if I can conclude what I want with the above.
Surprisingly, the space turns out to be metrizable. To see this, let $\delta_k \in c_{00}$ be the sequence which is one in the $k$-th entry and zero elsewhere. Now, note that if $(\varphi_\alpha)_{\alpha}$ is a net in $X^\ast$ and $\varphi \in X^\ast$, then $\varphi_\alpha \to \varphi$ in the weak-$\ast$ topology if and only if $\varphi_\alpha (\delta_n) \to \varphi(\delta_n)$ for all $n \in \Bbb{N}$. This uses that every $x \in X$ is a finite linear combination of $\delta_n$'s.
Now, define $$ d (\varphi, \psi) := \sum_{n=1}^\infty \min \{ 2^{-n}, |\varphi(\delta_n) - \psi(\delta_n)| \} \quad \text{for} \quad \varphi,\psi \in X^\ast . $$ It is not hard to see that this is a metric (change that each summand is a pseudo-metric) and that $d(\varphi_\alpha, \varphi) \to 0$ if and only if $\varphi_\alpha(\delta_n) \to \varphi(\delta_n)$ for all $n \in \Bbb{N}$.
This shows that the topology you are interested in is induced by the metric $d$.