I know that the spaces $W^{k,p}(\Omega)$ are separable and reflexive ($\Omega \subset \mathbb{R}^n$ open and bounded) for $p \in (1, \infty)$. I also learned that in a separable Banach space $X$ there exists a metric on the closed unit ball $ B \subset X^* $ which induces the weak* topology on $B$.
Now my thinking goes as follows: Since $W^{k,p}(\Omega)$ is separable, there exists a metric on the closed unit ball inducing the weak* topology. Can this metric be extended to all of $W^{k,p}(\Omega)$? Or at least to $W_0^{k,p}(\Omega)$? Since $W^{k,p}(\Omega)$ is also reflexive, the weak and weak* topologies coincide. Therefore, there exists a metric on the closed unit ball inducing the weak topology too. Extends this metric? I guess both induced metrics are isomorphic?
(My actual goal is to show that $W_0^{k,p}(\Omega)$ is first-countable.)
This does not have much to do with Sobolev spaces. Let thus $X$ be a separable Banach space with a countable dense set $S=\{x_n:n\in\mathbb N\}$. The topology on $X^*$ of pointwise convergence on $S$ is given by the increasing sequence of semi-norms $q_n(x^*)=\max\{|x^*(x_k)|:1\le k\le n\}$. This topology is Hausdorff (by the density of $S$) and metrizable, e.g., by the metric $$d(x^*,y^*)=\sup\{q_n(x^*-y^*) \wedge 1/n:n\in\mathbb N\}$$ where $a\wedge b=\min\{a,b\}$. By Alaoglu, the unit ball $B_{X^*}$ is $\sigma(X^*,X)$-compact and since a compact space does not have stricly coarser Hausdorff topologies $d$ induces the weak$^*$-topology on $B_{X^*}$.
However, for infinite dimensional Banach spaces, the weak$^*$ topology on the full dual $X^*$ is never metrizable: Otherwise, it would then be (sequentially) complete (because $\sigma(X^*,X)$-Cauchy sequences are pointwise bounded and hence uniformly bounded and the compactness of balls $rB_{X^*}$ then yields converging subsequences which implies convergence of the full Cauchy sequence) and the open mapping theorem (for Fréchet spaces) would imply that weak$^*$-topology coincides with the dual norm topology which is not true (because non of the semi-norms generating $\sigma(X^*,X)$ is a proper norm).