[proofplan] This proof is a sketch of the main steps; the full argument is the standard $L^2$-elliptic compactness theorem for the Hodge Laplacian on a closed Riemannian manifold and rests on three black boxes from elliptic PDE theory: (i) Garding's inequality for $\Delta$, which produces an a priori $H^1$-bound on $\alpha_n$ from the $L^2$-bounds on $\alpha_n$ and $\Delta\alpha_n$; (ii) the [Rellich-Kondrachov](/theorems/64) compact embedding $H^1 \hookrightarrow\hookrightarrow L^2$, which extracts an $L^2$-Cauchy subsequence from the bounded $H^1$-sequence; and (iii) elliptic regularity (via partitions of unity and local interior estimates), which controls $\alpha_n$ in $H^1$ over each chart of a finite cover. The compactness of $M$ enters in two places: it provides a finite atlas with uniform constants, and it makes the Rellich-Kondrachov compactness apply globally on $M$ rather than only on bounded subdomains. [/proofplan]

[step:Set up Sobolev spaces of $p$-forms on a closed Riemannian manifold]Let $(M, g)$ be a compact oriented Riemannian manifold of dimension $n$ without boundary. Fix a finite atlas $\{(U_i, \varphi_i)\}_{i=1}^N$ trivialising the bundle $\Lambda^p T^*M$ over each chart, together with a smooth partition of unity $\{\chi_i\}_{i=1}^N$ subordinate to the atlas. The compactness of $M$ guarantees the existence of such a finite atlas. For a smooth $p$-form $\eta \in \Omega^p(M)$, define the global $L^2$-norm \begin{align*} \|\eta\|_{L^2}^2 := \int_M (\eta, \eta)_g \, \omega_g, \end{align*} where $(\cdot, \cdot)_g$ is the fibrewise inner product on $\Lambda^p T^*M$ and $\omega_g$ is the Riemannian volume form. This is the norm written $\|\cdot\|$ in the statement of the theorem. Define the global $H^1$-norm by \begin{align*} \|\eta\|_{H^1}^2 := \|\eta\|_{L^2}^2 + \|\nabla\eta\|_{L^2}^2, \end{align*} where $\nabla$ is the Levi-Civita connection extended to forms and $\|\nabla\eta\|_{L^2}^2 := \int_M (\nabla\eta, \nabla\eta)_g \, \omega_g$ uses the fibrewise inner product on $T^*M \otimes \Lambda^p T^*M$ induced by $g$. The Sobolev space $H^1(\Lambda^p T^*M)$ is the completion of $\Omega^p(M)$ in this norm.[/step]

[guided]Before stating the inequality we must fix the function spaces. The hypothesis of the theorem mentions $\|\Delta\alpha_n\|$, which only makes sense once we have a global $L^2$-norm on forms; the conclusion mentions a Cauchy property, which makes sense only once we have a complete normed space. So we set up $L^2$ and $H^1$ on forms. *Why a finite atlas?* On a non-compact manifold one would have to deal with global versus local Sobolev spaces and worry about uniform constants. On a closed manifold the picture simplifies dramatically: compactness of $M$ produces a finite atlas $\{(U_i, \varphi_i)\}_{i=1}^N$ on each of whose charts the bundle $\Lambda^p T^*M$ trivialises, plus a smooth partition of unity $\{\chi_i\}_{i=1}^N$ subordinate to the atlas. Finiteness of $N$ is what will let later arguments — the diagonal extraction and the gluing of chart-wise Rellich-Kondrachov — terminate. *The $L^2$-norm.* Given a smooth $p$-form $\eta \in \Omega^p(M)$, the fibrewise inner product $(\cdot, \cdot)_g$ on $\Lambda^p T^*M$ — induced from $g$ — defines a smooth scalar function $(\eta, \eta)_g: M \to \mathbb{R}_{\ge 0}$. Integrating against the Riemannian volume form $\omega_g$ produces the global $L^2$-norm \begin{align*} \|\eta\|_{L^2}^2 := \int_M (\eta, \eta)_g \, \omega_g. \end{align*} This is the very norm denoted $\|\cdot\|$ in the statement of the theorem; the bounds $\|\alpha_n\| \le C$ and $\|\Delta\alpha_n\| \le C$ are bounds in this norm. *The $H^1$-norm.* To control derivatives of $\eta$, we use the Levi-Civita connection $\nabla$ extended in the canonical way to $p$-forms. Then $\nabla\eta$ is a section of $T^*M \otimes \Lambda^p T^*M$, on which $g$ induces a fibrewise inner product. The associated $L^2$-norm of the derivative is $\|\nabla\eta\|_{L^2}^2 := \int_M (\nabla\eta, \nabla\eta)_g \, \omega_g$, and the global $H^1$-norm is \begin{align*} \|\eta\|_{H^1}^2 := \|\eta\|_{L^2}^2 + \|\nabla\eta\|_{L^2}^2. \end{align*} The Sobolev space $H^1(\Lambda^p T^*M)$ is the completion of $\Omega^p(M)$ in this norm — completeness ensures we have a Hilbert space to which the Rellich-Kondrachov compactness will apply. *Equivalence with the chart-wise definition.* It is also possible — and sometimes more convenient — to define $\eta \in H^1$ by requiring that for each chart $(U_i, \varphi_i)$, the localised form $\chi_i\eta$, when transferred to $\varphi_i(U_i) \subseteq \mathbb{R}^n$ and viewed component-wise as a tuple of scalar functions, lies in the standard scalar Sobolev space $H^1(\varphi_i(U_i))$. The two definitions yield *equivalent* norms because the chart transitions are smooth and $M$ is compact, so the metric components $g_{ij}$, their derivatives, and the Christoffel symbols $\Gamma_{ij}^k$ are bounded uniformly. This equivalence is what allows the partition-of-unity argument below to translate global $H^1$-bounds into chart-wise $H^1_0$-bounds and back.[/guided]

[step:Apply Garding's inequality for $\Delta$ to bound $\|\alpha_n\|_{H^1}$]Garding's inequality for the Hodge Laplacian on $(M, g)$ states: there exist constants $C_1 = C_1(M, g) > 0$ and $C_2 = C_2(M, g) > 0$ such that for every $\eta \in \Omega^p(M)$, \begin{align*} \|\eta\|_{H^1}^2 \le C_1 \langle\langle \Delta\eta, \eta \rangle\rangle_g + C_2 \|\eta\|_{L^2}^2, \tag{G} \end{align*} where $\langle\langle \cdot, \cdot \rangle\rangle_g$ is the global $L^2$ inner product. The constants $C_1, C_2$ depend only on the geometry of $(M, g)$, not on $\eta$. The proof of (G) rests on uniform ellipticity of $\Delta$ (whose principal symbol is $|\xi|^2 \mathrm{Id}_{\Lambda^p}$) combined with a partition-of-unity argument over the finite atlas: locally, in normal coordinates, the leading-order part of $\Delta$ on $p$-forms is the standard scalar Laplacian acting component-wise, for which the analogous inequality is \begin{align*} \int |\nabla u|^2 \, d\mathcal{L}^n \le \int u(-\Delta_{\mathbb{R}^n} u) \, d\mathcal{L}^n + \text{lower-order terms}, \end{align*} plus controlled lower-order zeroth- and first-order error terms from the curvature of $(M, g)$. The full statement is a standard fact in elliptic PDE on compact manifolds. Apply Garding's inequality (G) to $\eta := \alpha_n$ and use Cauchy-Schwarz on the right: \begin{align*} \|\alpha_n\|_{H^1}^2 \le C_1 \langle\langle \Delta\alpha_n, \alpha_n \rangle\rangle_g + C_2 \|\alpha_n\|_{L^2}^2 \le C_1 \|\Delta\alpha_n\|_{L^2} \|\alpha_n\|_{L^2} + C_2 \|\alpha_n\|_{L^2}^2 \le C_1 C^2 + C_2 C^2. \end{align*} Thus \begin{align*} \|\alpha_n\|_{H^1} \le C', \tag{B} \end{align*} where $C' := \sqrt{(C_1 + C_2) C^2}$ depends only on $C$ and the geometry of $(M, g)$, not on $n$. The sequence $(\alpha_n)$ is therefore uniformly bounded in $H^1$.[/step]

[guided]The hypothesis of the theorem controls $\alpha_n$ and $\Delta\alpha_n$ in $L^2$. The Rellich-Kondrachov theorem we want to invoke in the next step requires a bound in $H^1$. The bridge between the two is Garding's inequality, an elliptic regularity statement: $\Delta$ is a second-order elliptic operator, so two $L^2$-bounds — one on $\alpha_n$ itself, one on its image under $\Delta$ — should produce a bound on the *first*-order norm $\|\alpha_n\|_{H^1}$. **Statement of Garding's inequality for the Hodge Laplacian.** Let $(M, g)$ be a compact Riemannian manifold without boundary. There exist constants $C_1 = C_1(M, g) > 0$ and $C_2 = C_2(M, g) > 0$ such that for every smooth $p$-form $\eta \in \Omega^p(M)$, \begin{align*} \|\eta\|_{H^1}^2 \le C_1 \langle\langle \Delta\eta, \eta \rangle\rangle_g + C_2 \|\eta\|_{L^2}^2, \tag{G} \end{align*} where $\langle\langle \cdot, \cdot \rangle\rangle_g$ is the global $L^2$ inner product of forms induced by $g$. The constants $C_1, C_2$ depend only on the geometry of $(M, g)$ — concretely, on uniform bounds for $g_{ij}$, $\partial g_{ij}$, $\Gamma_{ij}^k$, and the curvature $R_{ijkl}$ over the finite atlas, which are finite because $M$ is compact — but not on $\eta$. This is a standard result in elliptic PDE on compact Riemannian manifolds; we use it as a quoted theorem (see, e.g., Aubin, *Some Nonlinear Problems in Riemannian Geometry*, Chapter 3, or Warner, *Foundations of Differentiable Manifolds and Lie Groups*, Chapter 6). The proof of (G) rests on two ingredients: (i) Uniform ellipticity of $\Delta$. The principal symbol of the Hodge Laplacian is $|\xi|^2 \mathrm{Id}_{\Lambda^p}$, so $\Delta$ is a uniformly elliptic second-order operator on each chart of the finite atlas. (ii) Localisation via the partition of unity. In normal coordinates around any point, the leading-order part of $\Delta$ on $p$-forms is the standard scalar Laplacian $-\Delta_{\mathbb{R}^n}$ acting component-wise, for which the analogous Euclidean estimate \begin{align*} \int |\nabla u|^2 \, d\mathcal{L}^n \le \int u(-\Delta_{\mathbb{R}^n} u) \, d\mathcal{L}^n + \text{lower-order terms} \end{align*} follows from integration by parts. The chart-wise estimates glue (using compactness of $M$ for uniform constants) into the global statement (G), with controlled lower-order zeroth- and first-order error terms coming from the curvature of $(M, g)$. *Why Garding holds, conceptually.* On a closed manifold, integration by parts using the [Co-differential is Formal Adjoint of $d$](/theorems/2742) gives the identity \begin{align*} \langle\langle \Delta\alpha, \alpha \rangle\rangle_g = \|d\alpha\|_{L^2}^2 + \|\delta\alpha\|_{L^2}^2, \end{align*} so $\langle\langle \Delta\alpha, \alpha \rangle\rangle$ controls the *Hodge* energy. The non-trivial content of (G) is that this Hodge energy plus $\|\alpha\|_{L^2}^2$ is equivalent — via a Weitzenböck formula, with constants depending on curvature — to the *full* $H^1$-norm $\|\alpha\|_{L^2}^2 + \|\nabla\alpha\|_{L^2}^2$. That equivalence is what makes (G) an elliptic regularity statement rather than a tautology. **Applying (G) to $\alpha_n$.** Set $\eta := \alpha_n$ in (G): \begin{align*} \|\alpha_n\|_{H^1}^2 \le C_1 \langle\langle \Delta\alpha_n, \alpha_n \rangle\rangle_g + C_2 \|\alpha_n\|_{L^2}^2. \end{align*} The right-hand side mixes the $L^2$ inner product $\langle\langle \Delta\alpha_n, \alpha_n \rangle\rangle_g$ with the squared $L^2$-norm of $\alpha_n$. We bound the inner product using Cauchy-Schwarz in $L^2$ — this is allowed because $(\cdot, \cdot)_{L^2}$ is a genuine inner product, hence satisfies $|\langle\langle f, g \rangle\rangle| \le \|f\|_{L^2} \|g\|_{L^2}$. Combining with the hypothesis $\|\Delta\alpha_n\|_{L^2}, \|\alpha_n\|_{L^2} \le C$: \begin{align*} \|\alpha_n\|_{H^1}^2 \le C_1 \|\Delta\alpha_n\|_{L^2} \|\alpha_n\|_{L^2} + C_2 \|\alpha_n\|_{L^2}^2 \le C_1 C \cdot C + C_2 C^2 = (C_1 + C_2) C^2. \end{align*} Taking square roots gives the uniform $H^1$-bound \begin{align*} \|\alpha_n\|_{H^1} \le C', \tag{B} \end{align*} with $C' := \sqrt{(C_1 + C_2) C^2} = C\sqrt{C_1 + C_2}$. This constant depends only on $C$ (from the hypothesis) and the geometric constants $C_1, C_2$ of $(M, g)$ — crucially, *not* on $n$. The sequence $(\alpha_n)$ is therefore uniformly bounded in $H^1$, which is exactly the hypothesis the Rellich-Kondrachov theorem will need.[/guided]

[step:Apply the Rellich-Kondrachov compact embedding to extract an $L^2$-Cauchy subsequence]The continuous embedding \begin{align*} H^1(\Lambda^p T^*M) \hookrightarrow L^2(\Lambda^p T^*M) \end{align*} is in fact compact: the [Rellich-Kondrachov](/theorems/64) theorem (applied chart-wise via the partition of unity, with constants uniform over the finite atlas) gives that every bounded sequence in $H^1(\Lambda^p T^*M)$ has a subsequence convergent in $L^2(\Lambda^p T^*M)$. The compactness of $M$ is the reason the chart-wise compact embeddings glue to a global compact embedding without additional hypotheses on the support of the forms. By (B), $(\alpha_n)$ is bounded in $H^1$. Hence by the Rellich-Kondrachov compact embedding, there exist a subsequence $(\alpha_{n_k})$ and a form $\alpha \in L^2(\Lambda^p T^*M)$ such that \begin{align*} \alpha_{n_k} \to \alpha \text{ in } L^2(\Lambda^p T^*M) \text{ as } k \to \infty. \end{align*} In particular, $(\alpha_{n_k})$ is a Cauchy sequence in $L^2$: \begin{align*} \|\alpha_{n_k} - \alpha_{n_\ell}\|_{L^2} \le \|\alpha_{n_k} - \alpha\|_{L^2} + \|\alpha - \alpha_{n_\ell}\|_{L^2} \to 0 \quad \text{as } k, \ell \to \infty. \end{align*} This is the conclusion of the theorem.[/step]

[guided]We have a sequence $(\alpha_n)$ uniformly bounded in $H^1$ by (B). To extract an $L^2$-Cauchy subsequence, we want to upgrade the *continuous* embedding \begin{align*} H^1(\Lambda^p T^*M) \hookrightarrow L^2(\Lambda^p T^*M) \end{align*} — which is automatic from the definition $\|\eta\|_{L^2}^2 \le \|\eta\|_{H^1}^2$ and which alone only gives weak compactness of bounded $H^1$-sequences in $L^2$ — to a *compact* embedding, which gives strong $L^2$-convergence of a subsequence. This is precisely what the [Rellich-Kondrachov](/theorems/64) theorem provides on bounded Euclidean domains. The work of this step is to transfer that Euclidean compactness to the compact manifold $M$ using partition of unity, performing the gluing argument explicitly. **Step (a) — Localise via the partition of unity.** Recall the smooth partition of unity $\{\chi_i\}_{i=1}^N$ subordinate to the atlas $\{(U_i, \varphi_i)\}_{i=1}^N$, satisfying $\sum_{i=1}^N \chi_i \equiv 1$. Each localised form $\chi_i\alpha_n$ is supported in $U_i$. Transferring via the chart $\varphi_i$ to $V_i := \varphi_i(U_i) \subseteq \mathbb{R}^n$ produces a tuple of compactly supported scalar functions on $V_i$ — i.e., elements of $H^1_0(V_i)$. Using that the chart trivialises the bundle and that the metric components are uniformly bounded, the chain rule gives \begin{align*} \|\chi_i\alpha_n\|_{H^1(V_i)} \le K_i \|\alpha_n\|_{H^1(M)} \le K_i C', \end{align*} where $K_i < \infty$ depends on uniform bounds for $\chi_i$, $\nabla\chi_i$, the chart metric components, and the Christoffel symbols (all finite by compactness of $M$). The crucial point: $K_i$ is independent of $n$. **Step (b) — Diagonal extraction across $N$ charts.** We have, for each $i \in \{1, \ldots, N\}$, a sequence $(\chi_i\alpha_n)_n$ uniformly bounded in $H^1_0(V_i)$. By [Rellich-Kondrachov](/theorems/64) on the bounded open set $V_i$ — whose hypotheses (boundedness of $V_i$, $H^1_0$ functions, embedding target $L^2$) are all met — we may extract an $L^2(V_i)$-convergent subsequence. We extract diagonally. Apply Rellich-Kondrachov to $(\chi_1\alpha_n)_n$ to obtain a subsequence $S_1 \subseteq \mathbb{N}$ with $(\chi_1\alpha_n)_{n \in S_1}$ convergent in $L^2(V_1)$. Apply again to $(\chi_2\alpha_n)_{n \in S_1}$ — still bounded in $H^1_0(V_2)$ — to extract $S_2 \subseteq S_1$ with $(\chi_2\alpha_n)_{n \in S_2}$ convergent in $L^2(V_2)$ (and still convergent in $L^2(V_1)$, since $S_2 \subseteq S_1$). Iterate $N$ times to obtain a chain $S_N \subseteq S_{N-1} \subseteq \cdots \subseteq S_1$ such that $(\chi_i\alpha_n)_{n \in S_N}$ converges in $L^2(V_i)$ for *every* $i$ simultaneously. Re-index $S_N$ as $(\alpha_{n_k})_k$. Finiteness of the atlas — itself a consequence of compactness of $M$ — is what terminates the extraction; on a non-compact manifold one would need a more delicate countable diagonal argument. **Step (c) — Glue chart-wise convergence to global $L^2$-convergence.** Using $\sum_i \chi_i \equiv 1$ on $M$, decompose the difference \begin{align*} \alpha_{n_k} - \alpha_{n_\ell} = \sum_{i=1}^N (\chi_i \alpha_{n_k} - \chi_i \alpha_{n_\ell}). \end{align*} This is a genuine decomposition with weights summing to $1$ — *not* an overlapping cover argument that would double-count contributions in chart intersections. Apply the triangle inequality in $L^2(M)$: \begin{align*} \|\alpha_{n_k} - \alpha_{n_\ell}\|_{L^2(M)} \le \sum_{i=1}^N \|\chi_i \alpha_{n_k} - \chi_i \alpha_{n_\ell}\|_{L^2(M)}. \end{align*} Each term on the right is — up to a fixed Jacobian factor coming from $\det D\varphi_i^{-1}$ and the chart density of $\omega_g$, both bounded uniformly because $M$ is compact — equal to $\|\chi_i\alpha_{n_k} - \chi_i\alpha_{n_\ell}\|_{L^2(V_i)}$. By Step (b), $(\chi_i\alpha_{n_k})_k$ converges in $L^2(V_i)$ and is therefore Cauchy in $L^2(V_i)$, so this chart-wise difference tends to $0$ as $k, \ell \to \infty$. The right-hand side is a *finite* sum of $N$ such terms — finiteness once again from compactness of $M$ — so \begin{align*} \|\alpha_{n_k} - \alpha_{n_\ell}\|_{L^2(M)} \le \sum_{i=1}^N \|\chi_i \alpha_{n_k} - \chi_i \alpha_{n_\ell}\|_{L^2(M)} \to 0 \quad \text{as } k, \ell \to \infty. \end{align*} Hence $(\alpha_{n_k})$ is Cauchy in $L^2(\Lambda^p T^*M)$. By completeness of $L^2$ — this is where the choice of the *complete* normed space matters — there exists a limit $\alpha \in L^2(\Lambda^p T^*M)$ with $\alpha_{n_k} \to \alpha$ in $L^2$. *Why a partition of unity instead of a bare cover?* A bare cover would give chart-wise convergence but no canonical way to combine chart-wise estimates into a global one — overlaps would be double-counted, and one would have to work harder to deduce a global Cauchy property. The partition of unity provides a genuine *decomposition* $\alpha_n = \sum_i \chi_i\alpha_n$ with weights summing to $1$, so the triangle inequality in Step (c) becomes a finite sum of chart-wise terms, each controlled by Rellich-Kondrachov on its individual chart.[/guided]

[step:Conclude] We have produced a subsequence $(\alpha_{n_k})$ that is Cauchy in $L^2(\Lambda^p T^*M)$, equivalently Cauchy with respect to the norm $\|\cdot\|$ in the statement of the theorem. This is the desired conclusion. [/step]