The zero-boundary W1p space is separable and reflexive

Let $D$ be the countable rational box-step dense family in [F4]. Zero extension embeds $L^p(\mathrm{\Omega })$ isometrically in $L^p({\mathbb{R}}^d)$. Given $g\in L^p(\mathrm{\Omega })$, choose $s\in D$ approximating its zero extension $\tilde{g}$; then $\mathrm{\parallel }s{\mathrm{\mid }}_{\mathrm{\Omega }}-g{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}\le \mathrm{\parallel }s-\tilde{g}{\mathrm{\parallel }}_{L^p({\mathbb{R}}^d)}$. Thus the restrictions of $D$ are countable and dense in $L^p(\mathrm{\Omega })$. Their finite products give a countable dense subset of $Y$; balls centered there with rational radii form a countable base by [F12].

Let $(u_j)$ be Cauchy in $W^{1,p}(\mathrm{\Omega })$. By [F1], the coordinate sequences $(u_j)$ and $({\mathrm{\partial }}_i{u}_j)$ are Cauchy in $L^p(\mathrm{\Omega })$ for each $i$. Countable Choice is available by [A1], so [F6] gives limits $u,g_i\in L^p(\mathrm{\Omega })$. Fix $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ with compact support $K$. By [F10], $K$ has finite measure and $\phi ,{\mathrm{\partial }}_i\phi$ are bounded there; Hölder [F9] therefore lets both sides of ${\int }_{\mathrm{\Omega }}{u}_j{\mathrm{\partial }}_i\phi =-{\int }_{\mathrm{\Omega }}({\mathrm{\partial }}_i{u}_j)\phi$ pass to the limits $u$ and $g_i$. Hence $g_i={\mathrm{\partial }}_{i}u$ weakly for each $i$, so $u\in W^{1,p}(\mathrm{\Omega })$ and $u_j\to u$ in its norm by coordinate convergence. Thus $W^{1,p}(\mathrm{\Omega })$ is Banach.

Each factor of $Y$ is Banach by [F6], so the finite product is Banach for its maximum norm by [F7]. Let $(y_j)$ be a bounded sequence in $Y$. Its finitely many coordinate sequences are bounded in $L^p(\mathrm{\Omega })$; by [F8] and the bounded-sequence criterion [F11], successive subsequence extractions give one subsequence weakly convergent in every coordinate. Every bounded linear functional on $Y$ is the sum of its restrictions to the coordinate factors, so this subsequence converges weakly in $Y$. Thus [F11] makes $Y$ reflexive. By [F1], $\mathrm{\Phi }$ is a bounded linear isomorphism onto its image $S:=\mathrm{\Phi }(W^{1,p}(\mathrm{\Omega }))$. The image $S$ is closed in $Y$: a convergent sequence in $S$ has Cauchy preimages by the lower norm bound in [F1], and they converge by step 1.2. Hahn–Banach and [F3] make $S$ reflexive. The inverse isomorphism ${\mathrm{\Phi }}^{-1}:S\to W^{1,p}(\mathrm{\Omega })$ preserves weak convergence, so [F11] shows that $W^{1,p}(\mathrm{\Omega })$ is reflexive.

By [F2], $W_0^{1,p}(\mathrm{\Omega })$ is a closed linear subspace of the reflexive Banach space $W^{1,p}(\mathrm{\Omega })$ from step 2.1. Hahn–Banach and [F3] therefore make $W_0^{1,p}(\mathrm{\Omega })$ reflexive and Banach with its inherited norm.

Let $S_0=\mathrm{\Phi }(W_0^{1,p}(\mathrm{\Omega }))\subseteq Y$. The countable base from step 1.1 restricts to a countable base of $S_0$. By [A1], Countable Choice selects one point from each nonempty member of this relative base; the resulting countable set meets every nonempty relatively open set and is dense in $S_0$. Since $\mathrm{\Phi }$ is an isomorphism with equivalent norms by [F1], $W_0^{1,p}(\mathrm{\Omega })$ is separable by [F5].

Steps 3.1 and 4.1 establish that $W_0^{1,p}(\mathrm{\Omega };\mathbb{R})$ is a separable reflexive real Banach space with its inherited norm, as claimed.