[proofplan] The argument is a direct application of the [Sobolev Space is Banach](/theorems/3094) result, which establishes that $W^{1,p}(\Omega)$ is reflexive for $1 < p < \infty$. By the Banach-Alaoglu theorem in its sequential form, every bounded sequence in a reflexive Banach space has a weakly convergent subsequence. We translate this abstract weak convergence in $W^{1,p}$ into the equivalent component-wise statement: the sequence converges weakly in $L^p(\Omega)$ and each weak partial derivative converges weakly in $L^p(\Omega)$. The translation uses the isometric embedding $u \mapsto (u, \nabla u)$ of $W^{1,p}(\Omega)$ into $L^p(\Omega)^{n+1}$, which identifies weak convergence in the Sobolev space with component-wise weak convergence in $L^p$. The reflexivity hypothesis is essential: for $p = 1$ or $p = \infty$, the space fails to be reflexive and only weak-$*$ subsequential limits exist (not addressed by this theorem). [/proofplan]

[step:Invoke reflexivity of $W^{1,p}(\Omega)$ for $1 < p < \infty$] By the [Sobolev Space is Banach](/theorems/3094) theorem, $W^{1,p}(\Omega)$ is a reflexive Banach space whenever $1 < p < \infty$ and $\Omega \subset \mathbb{R}^n$ is open. The hypotheses required by Theorem 3094 — that $\Omega$ is open and $1 < p < \infty$ — are exactly those given in the present theorem (boundedness of $\Omega$ is an additional assumption here but is not needed for reflexivity; reflexivity holds for all open $\Omega$). Recall the isometric linear embedding underlying the reflexivity argument: \begin{align*} \iota: W^{1,p}(\Omega) &\to L^p(\Omega)^{n+1} \\ u &\mapsto \iota(u) := (u, \partial_{x_1} u, \dots, \partial_{x_n} u), \end{align*} where $L^p(\Omega)^{n+1}$ is the product Banach space equipped with the norm $\|(f_0, f_1, \dots, f_n)\|_{L^p(\Omega)^{n+1}}^p := \sum_{i=0}^n \|f_i\|_{L^p(\Omega)}^p$. This map satisfies $\|\iota(u)\|_{L^p(\Omega)^{n+1}} = \|u\|_{W^{1,p}(\Omega)}$ by the very definition of the $W^{1,p}$ norm (with the convention that the $W^{1,p}$ norm is the $\ell^p$ combination of the $L^p$ norms of $u$ and its partials, modulo the choice of equivalent norm). The image $\iota(W^{1,p}(\Omega))$ is a closed linear subspace of $L^p(\Omega)^{n+1}$: indeed, this is precisely the content of the completeness step in Theorem 3094, where a Cauchy sequence $(\iota(u_k))$ in $L^p(\Omega)^{n+1}$ converges to a vector of the form $\iota(u)$ for some $u \in W^{1,p}(\Omega)$. [/step]

[step:Apply Banach-Alaoglu (sequential reflexive form) to extract a weakly convergent subsequence]By hypothesis, $(u_k)_{k \ge 1}$ is bounded in $W^{1,p}(\Omega)$: there exists $M > 0$ such that \begin{align*} \|u_k\|_{W^{1,p}(\Omega)} \le M \quad \text{for all } k \ge 1. \end{align*} The Eberlein-Šmulian theorem states that every bounded sequence in a reflexive Banach space has a weakly convergent subsequence. (Equivalently: closed bounded sets in a reflexive Banach space are weakly sequentially compact.) Since $W^{1,p}(\Omega)$ is reflexive for $1 < p < \infty$ by the previous step, there exist a subsequence $(u_{k_j})_{j \ge 1}$ and a function $u \in W^{1,p}(\Omega)$ such that \begin{align*} u_{k_j} \rightharpoonup u \quad \text{weakly in } W^{1,p}(\Omega) \text{ as } j \to \infty. \end{align*} By definition, this means: for every continuous linear functional $F \in (W^{1,p}(\Omega))^*$, \begin{align*} F(u_{k_j}) \to F(u) \quad \text{as } j \to \infty. \end{align*}[/step]

[guided]**Why reflexivity matters.** The classical Heine-Borel theorem fails in infinite-dimensional Banach spaces: closed bounded sets are no longer norm-compact. The right substitute is *weak* compactness, and the Eberlein-Šmulian theorem is the bridge: it asserts that in a reflexive Banach space $X$, the closed unit ball $\overline{B}_X$ is weakly sequentially compact — every sequence in $\overline{B}_X$ has a weakly convergent subsequence with limit in $\overline{B}_X$. By scaling, every bounded sequence in $X$ has a weakly convergent subsequence. The proof of Eberlein-Šmulian for separable reflexive spaces is constructive and uses the Banach-Alaoglu theorem on the dual: the unit ball of $X^*$ is weak-$*$ compact, and via the canonical isometric isomorphism $X \cong X^{**}$ (which is what reflexivity *means*), bounded sequences in $X$ correspond to bounded sequences in $X^{**}$ that admit weak-$*$ convergent subsequences, which translate back to weakly convergent subsequences in $X$. For non-separable reflexive spaces, the argument is more delicate but the statement remains. **Hypothesis verification.** Theorem 3094 gives reflexivity of $W^{1,p}(\Omega)$ exactly under the hypothesis $1 < p < \infty$. The statement of the present theorem also has $1 < p < \infty$, so reflexivity applies. The boundedness of $\Omega$ is **not** required for reflexivity, but is part of the present theorem's hypotheses for compatibility with downstream applications. **The bound.** The hypothesis that $(u_k)$ is bounded in $W^{1,p}$ is the crucial ingredient: bounded $+$ reflexive $\Rightarrow$ weakly subsequentially compact. We extract a subsequence $(u_{k_j})$ and a limit $u \in W^{1,p}(\Omega)$ with $u_{k_j} \rightharpoonup u$ weakly in $W^{1,p}$. **Definition of weak convergence.** Recall that weak convergence in a Banach space $X$ is convergence in the weak topology, equivalently: $F(u_{k_j}) \to F(u)$ for every $F \in X^*$. So at this stage we have \begin{align*} \forall F \in (W^{1,p}(\Omega))^*: \quad F(u_{k_j}) \to F(u). \end{align*} The remaining work is to translate this abstract statement into the concrete component-wise statement claimed by the theorem.[/guided]

[step:Translate weak convergence in $W^{1,p}$ to component-wise weak convergence in $L^p$]We prove that $u_{k_j} \rightharpoonup u$ weakly in $W^{1,p}(\Omega)$ is equivalent to: \begin{align*} u_{k_j} \rightharpoonup u \quad \text{weakly in } L^p(\Omega), \quad \text{and} \quad \partial_{x_i} u_{k_j} \rightharpoonup \partial_{x_i} u \quad \text{weakly in } L^p(\Omega) \text{ for } i = 1, \dots, n. \end{align*} **Direction 1: weak in $W^{1,p}$ implies component-wise weak in $L^p$.** Let $g \in L^{p'}(\Omega)$ where $p' = p/(p-1)$ is the Hölder conjugate of $p$. Define the linear functional \begin{align*} F_g: W^{1,p}(\Omega) &\to \mathbb{R} \\ v &\mapsto F_g(v) := \int_\Omega v\, g\, d\mathcal{L}^n. \end{align*} By Hölder's inequality with exponents $(p, p')$, \begin{align*} |F_g(v)| \le \|v\|_{L^p(\Omega)}\|g\|_{L^{p'}(\Omega)} \le \|v\|_{W^{1,p}(\Omega)}\|g\|_{L^{p'}(\Omega)}, \end{align*} since $\|v\|_{L^p} \le \|v\|_{W^{1,p}}$ by definition of the $W^{1,p}$ norm. So $F_g \in (W^{1,p}(\Omega))^*$. Weak convergence $u_{k_j} \rightharpoonup u$ in $W^{1,p}$ then gives $F_g(u_{k_j}) \to F_g(u)$, i.e. \begin{align*} \int_\Omega u_{k_j}\, g\, d\mathcal{L}^n \to \int_\Omega u\, g\, d\mathcal{L}^n \quad \text{for every } g \in L^{p'}(\Omega). \end{align*} This is precisely the statement that $u_{k_j} \rightharpoonup u$ weakly in $L^p(\Omega)$ (using the standard duality $(L^p)^* = L^{p'}$ for $1 < p < \infty$ via the Riesz representation theorem for $L^p$ spaces). Similarly, fix $i \in \{1, \dots, n\}$ and define \begin{align*} G_{g, i}: W^{1,p}(\Omega) &\to \mathbb{R} \\ v &\mapsto G_{g, i}(v) := \int_\Omega \partial_{x_i} v\, g\, d\mathcal{L}^n. \end{align*} Hölder again yields $|G_{g, i}(v)| \le \|\partial_{x_i} v\|_{L^p}\|g\|_{L^{p'}} \le \|v\|_{W^{1,p}}\|g\|_{L^{p'}}$, so $G_{g, i} \in (W^{1,p}(\Omega))^*$. Weak convergence in $W^{1,p}$ gives $G_{g,i}(u_{k_j}) \to G_{g,i}(u)$, i.e. \begin{align*} \int_\Omega \partial_{x_i} u_{k_j}\, g\, d\mathcal{L}^n \to \int_\Omega \partial_{x_i} u\, g\, d\mathcal{L}^n \quad \text{for every } g \in L^{p'}(\Omega), \end{align*} which is $\partial_{x_i} u_{k_j} \rightharpoonup \partial_{x_i} u$ weakly in $L^p(\Omega)$. **Direction 2 (not needed for this theorem, but recorded for completeness): component-wise weak in $L^p$ implies weak in $W^{1,p}$.** This direction follows by representing every $F \in (W^{1,p}(\Omega))^*$ via the embedding $\iota$: by the Hahn-Banach theorem, every continuous linear functional on a closed subspace of $L^p(\Omega)^{n+1}$ extends to a continuous linear functional on the whole product space, which has dual $(L^p)^{(n+1)\,*} = (L^{p'})^{n+1}$. So every $F \in (W^{1,p})^*$ has the form $F(v) = \int_\Omega v g_0\, d\mathcal{L}^n + \sum_{i=1}^n \int_\Omega \partial_{x_i} v\, g_i\, d\mathcal{L}^n$ for some $(g_0, g_1, \dots, g_n) \in (L^{p'}(\Omega))^{n+1}$. Component-wise weak convergence then yields $F(u_{k_j}) \to F(u)$, hence weak convergence in $W^{1,p}$.[/step]

[guided]**Goal of the translation.** The Eberlein-Šmulian extraction in the previous step produced a subsequence with $u_{k_j} \rightharpoonup u$ weakly in $W^{1,p}(\Omega)$. The theorem statement, however, asks for a more concrete description: weak convergence in $L^p$ of the function and of each weak partial derivative. We need to show these two formulations are equivalent — and in particular, that the Sobolev weak convergence implies the component-wise $L^p$ weak convergence. **Embedding into $L^p$ via the natural inclusion.** The natural inclusion $W^{1,p}(\Omega) \hookrightarrow L^p(\Omega)$ is continuous: $\|v\|_{L^p(\Omega)} \le \|v\|_{W^{1,p}(\Omega)}$ (this is part of the definition of the Sobolev norm — the $L^p$ component is one of the terms in the sum). A continuous linear map between Banach spaces is automatically weakly continuous: if $v_j \rightharpoonup v$ in $W^{1,p}$ and $T: W^{1,p} \to L^p$ is the inclusion, then $T(v_j) \rightharpoonup T(v)$ in $L^p$. So weak convergence in $W^{1,p}$ implies weak convergence of the **values** in $L^p$. But we want a self-contained verification, not a black-box appeal. Here is the direct argument: for each $g \in L^{p'}(\Omega)$, the integration-against-$g$ functional \begin{align*} F_g: W^{1,p}(\Omega) &\to \mathbb{R}, & v &\mapsto \int_\Omega v\, g\, d\mathcal{L}^n \end{align*} is bounded on $W^{1,p}$ — Hölder's inequality on $L^p \times L^{p'}$ gives $|F_g(v)| \le \|v\|_{L^p}\|g\|_{L^{p'}} \le \|v\|_{W^{1,p}}\|g\|_{L^{p'}}$. Hence $F_g \in (W^{1,p})^*$. Weak convergence $u_{k_j} \rightharpoonup u$ in $W^{1,p}$ then gives $F_g(u_{k_j}) \to F_g(u)$, i.e. \begin{align*} \int_\Omega u_{k_j} g\, d\mathcal{L}^n \to \int_\Omega u\, g\, d\mathcal{L}^n. \end{align*} This is the very definition of weak convergence in $L^p(\Omega)$, recalling that $(L^p)^* = L^{p'}$ via $g \mapsto F_g$ (Riesz representation for $L^p$, $1 < p < \infty$). **The same trick for the partials.** The map $\partial_{x_i}: W^{1,p}(\Omega) \to L^p(\Omega)$ that takes $v$ to its weak $i$-th partial derivative is also continuous: $\|\partial_{x_i} v\|_{L^p} \le \|v\|_{W^{1,p}}$ by definition of the Sobolev norm. So $\partial_{x_i}$ is weakly continuous, and weak convergence in $W^{1,p}$ implies weak convergence of the partials in $L^p$. Concretely, for each $g \in L^{p'}(\Omega)$ and each $i$, the functional \begin{align*} G_{g, i}: W^{1,p}(\Omega) &\to \mathbb{R}, & v &\mapsto \int_\Omega \partial_{x_i} v\, g\, d\mathcal{L}^n \end{align*} is bounded on $W^{1,p}$, hence in the dual, hence weak convergence in $W^{1,p}$ gives convergence of $G_{g,i}(u_{k_j}) \to G_{g, i}(u)$, which is weak convergence of the partials in $L^p$. **Putting it together.** We have \begin{align*} u_{k_j} &\rightharpoonup u \quad \text{weakly in } L^p(\Omega), \\ \partial_{x_i} u_{k_j} &\rightharpoonup \partial_{x_i} u \quad \text{weakly in } L^p(\Omega), \quad i = 1, \dots, n. \end{align*} **On multi-index notation $D^\alpha$ with $|\alpha| \le 1$.** The theorem statement uses the multi-index notation $D^\alpha u_{k_j} \rightharpoonup D^\alpha u$ for $|\alpha| \le 1$. The case $\alpha = (0, \dots, 0)$ gives $D^\alpha = \mathrm{id}$, hence $D^\alpha u_{k_j} = u_{k_j}$ — and the convergence is weak in $L^p$. The cases $|\alpha| = 1$ are exactly the standard partial derivatives $\partial_{x_i}$ for $i = 1, \dots, n$. We have established weak $L^p$ convergence in each of these $n + 1$ cases, which is what the theorem asks for.[/guided]

[step:Conclude] Combining the previous steps: starting from a bounded sequence $(u_k)$ in $W^{1,p}(\Omega)$ with $1 < p < \infty$, we extracted a subsequence $(u_{k_j})$ and a limit $u \in W^{1,p}(\Omega)$ such that $u_{k_j} \rightharpoonup u$ weakly in $W^{1,p}(\Omega)$. The translation step showed this is equivalent to \begin{align*} u_{k_j} \rightharpoonup u \text{ weakly in } L^p(\Omega), \quad D^\alpha u_{k_j} \rightharpoonup D^\alpha u \text{ weakly in } L^p(\Omega)\ \text{for each multi-index } |\alpha| \le 1. \end{align*} This is precisely the conclusion of the theorem. The proof is complete. [/step]