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Compact embedding

In functional analysis, a compact embedding occurs when one Banach space X is continuously included in another Banach space Y such that the inclusion operator I: X \to Y, defined by Iu = u, is a compact operator.^[1] This means that bounded sets in X are mapped to precompact sets in Y, implying that every bounded sequence in X has a subsequence that converges in the norm of Y.^[1] A fundamental result establishing compact embeddings is the Rellich–Kondrachov theorem, which applies to Sobolev spaces on bounded domains in \mathbb{R}^n.^[1] Specifically, if \Omega \subset \mathbb{R}^n is an open, bounded set with C^1 boundary, then for $1 \leq p < \infty and $1 \leq q < p^* (where p^* = np/(n-p) is the Sobolev conjugate exponent if p < n, or any q if p \geq n), the Sobolev space W^{1,p}(\Omega) is compactly embedded in L^q(\Omega).^[1] This theorem extends to higher-order Sobolev spaces W^{k,p}(\Omega) under analogous conditions on the exponents and domain regularity, ensuring compactness when the target space has lower regularity.^[1] Compact embeddings play a central role in the theory of partial differential equations (PDEs), particularly in variational methods for establishing the existence and regularity of solutions.^[2] For instance, they enable the application of the direct method in the calculus of variations by guaranteeing that minimizing sequences converge strongly in weaker norms, thus yielding weak solutions to elliptic boundary value problems.^[2] Recent extensions include compact embeddings for variable exponent Sobolev spaces and on unbounded domains, which address more complex nonlinear PDEs such as semilinear elliptic equations.^[3]^[4]

Topological Setting

Definition

In topology, the notion of compact embedding is not as standardized as in functional analysis, but it can be understood in terms of subsets of topological spaces. A subset V of a topological space Y is said to be compactly embedded if there exists a compact subset K of Y such that V \subseteq K. More precisely, in some contexts, V is compactly embedded in an open set W \supseteq V if \overline{V} \subseteq \operatorname{Int}(W) and \overline{V} is compact, where \overline{V} denotes the closure of V. An example is a bounded closed subset of \mathbb{R}^n, such as the closed unit ball, whose closure is compact by the Heine-Borel theorem. This topological notion provides the foundation for the analytic concept of compact operators in normed spaces.

Properties

Compact embeddings preserve compactness: the continuous image of a compact set is compact. For an inclusion of a compact subset K \subseteq Y, K is compact in the subspace topology. In Hausdorff spaces, compact subsets are closed, so the inclusion of a compact subset is a closed embedding. In non-Hausdorff spaces, compact subsets need not be closed. For instance, in the indiscrete topology on a set with at least two points, every nonempty subset is compact but proper nonempty subsets are not closed.^[5] Sequential compactness is preserved under continuous maps: the continuous image of a sequentially compact set is sequentially compact.^[6] A classic example is the embedding of the circle S^1 into \mathbb{R}^2, where S^1 is compact and closed. Embeddings preserve intrinsic topological properties such as connectedness and path-connectedness, as the image is homeomorphic to the domain. Refinements of these properties appear in normed space settings, as discussed in later sections.

Analytic Setting

In Normed Spaces

In the analytic setting of normed spaces, the concept of compact embedding builds upon the topological notion of compactness by incorporating the norm structure to define relative compactness of images of bounded sets. A linear embedding i: E \to F between normed spaces E and F is compact if it is continuous and maps the closed unit ball B_E(0,1) = \{x \in E : \|x\|_E \leq 1\} of E to a relatively compact subset of F, meaning the closure of i(B_E(0,1)) is compact in F.^[7] This condition ensures that the embedding "improves" the regularity in a way that bounded sets in the domain become precompact in the codomain under the weaker topology induced by the norm on F.^[8] An equivalent sequential characterization of compactness for such an embedding is that for every bounded sequence \{x_n\} in E, the sequence \{i(x_n)\} in F admits a subsequence that converges in the norm of F.^[7] Compact embeddings are precisely the compact operators between normed spaces that are embeddings, i.e., continuous linear injections. In finite-dimensional normed spaces, every linear embedding is compact, as the unit ball is itself compact by the Heine-Borel theorem.^[9] However, in infinite-dimensional spaces, this fails in general; for instance, the identity operator on the Hilbert space \ell^2 is continuous but not compact, as the images of the standard orthonormal basis under the identity form a sequence with no convergent subsequence.^[8]

In Banach Spaces

In Banach spaces, compact embeddings refine the notion from normed spaces by leveraging completeness. Specifically, an embedding i: E \to F between Banach spaces E and F is compact if it is a bounded linear operator and the image of the closed unit ball of E under i is relatively compact in F; due to the completeness of F, this relative compactness is equivalent to the image being totally bounded.^[10] Such embeddings ensure that bounded sequences in E have subsequences converging in F, providing a bridge between the topologies induced by the norms of E and F. If the embedding is topological on its range—meaning the norms of E and F are equivalent there—it preserves more structure while maintaining compactness.^[10] A key property of compact embeddings in this setting is that the image of the unit ball is totally bounded, allowing coverage by finitely many balls of arbitrary small radius in the norm of F.^[10] Moreover, compact operators, including embeddings, admit approximation by finite-rank operators in the operator norm; that is, there exists a sequence of finite-rank operators converging to the embedding uniformly on bounded sets.^[10] For the adjoint, if i: E \to F is a compact embedding, then its adjoint i^*: F^* \to E^* is also compact, preserving the approximation properties in the dual spaces.^[10] Spectral implications further distinguish compact embeddings in Banach spaces. The spectrum of a compact operator T on a Banach space excludes any non-degenerate interval around non-zero points; more precisely, the non-zero part of the spectrum consists solely of at most countably many eigenvalues of finite geometric multiplicity, accumulating only at zero if infinite in number.^[11] This discrete nature arises from the Fredholm properties of compact perturbations of the identity, ensuring that T - \lambda I is Fredholm for \lambda \neq 0 with finite-dimensional kernel.^[11] An illustrative example is the embedding of the Sobolev space H^1(\mathbb{T}) into L^2(\mathbb{T}) on the one-dimensional torus \mathbb{T}, realized via Fourier series. Here, functions in H^1(\mathbb{T}) have Fourier coefficients \hat{f}(k) satisfying \sum_{k \in \mathbb{Z}} (1 + k^2) |\hat{f}(k)|^2 < \infty, and the embedding operator maps to L^2(\mathbb{T}) with norm \left( \sum_{k \in \mathbb{Z}} |\hat{f}(k)|^2 \right)^{1/2}. Compactness follows from the eigenvalue decay of the Laplacian on \mathbb{T}, where eigenvalues are -k^2 for k \in \mathbb{Z}, enabling approximation by finite-rank projections onto low-frequency modes.

Characterizations

General Criteria

In the context of normed spaces, a continuous embedding i: X \to Y (or inclusion) is compact if i is a compact operator, meaning that the image of every bounded set in X is precompact in Y. A basic result from topology is that if X is compact and i is continuous, then i(X) is compact, as the continuous image of a compact space is compact. In metric spaces, for subsets of \mathbb{R}^n, the Heine–Borel theorem characterizes compact subsets as closed and bounded, which can inform conditions under which embeddings preserve compactness for compact X. In metric spaces, the image of a totally bounded set under a uniformly continuous map is totally bounded; if the target space Y is complete, the closure of the image is compact. For compact embeddings, this applies to the images of bounded sets in X. In the analytic setting, particularly for L^p spaces, the Kolmogorov–Riesz criterion provides a necessary and sufficient condition for relative compactness of subsets, which applies to embeddings: a bounded set in L^p(\mathbb{R}^d) (for $1 \leq p < \infty) is relatively compact if it is uniformly bounded, has uniformly small tails (i.e., for every \varepsilon > 0, there exists R > 0 such that \int_{|x|>R} |f(x)|^p \, dx < \varepsilon^p for all f in the set), and is equicontinuous under translations (i.e., for every \varepsilon > 0, there exists \rho > 0 such that \int_{\mathbb{R}^d} |f(x+y) - f(x)|^p \, dx < \varepsilon^p for all f in the set and |y| < \rho). This criterion ensures that embeddings, such as inclusions between L^p spaces over \mathbb{R}^d, are compact when these uniform control conditions hold on bounded sets.^[12] Embeddings between spaces of continuous functions, such as from C(\overline{\Omega}) to L^p(\Omega) for $1 \leq p < \infty, are compact when the domain \Omega \subset \mathbb{R}^n is bounded and satisfies certain boundary conditions, like being Lipschitz or of class C^1, ensuring relative compactness of bounded sets in the target space.^[13] As a counterexample, the identity embedding of \mathbb{R} into itself is continuous but not compact, since the unit ball is not precompact (e.g., the sequence of characteristic functions of [n, n+1] has no convergent subsequence in L^p).

Role of Ascoli–Arzelà Theorem

The Ascoli–Arzelà theorem provides a fundamental criterion for relative compactness in the space C(K) of continuous real-valued functions on a compact metric space K, endowed with the supremum norm \|\cdot\|_\infty. Specifically, a subset F \subset C(K) is relatively compact if and only if it is bounded (i.e., \sup_{f \in F} \|f\|_\infty < \infty) and equicontinuous (i.e., for every \varepsilon > 0, there exists \delta > 0 such that |f(x) - f(y)| < \varepsilon whenever d(x,y) < \delta, for all f \in F).^[14] In the context of compact embeddings, the theorem is applied to verify compactness of inclusions into C(K) by showing that the image of the unit ball from the source space satisfies these conditions. If the embedding X \hookrightarrow C(K) maps the unit ball B_X of a Banach space X to a set that is bounded in \|\cdot\|_\infty and equicontinuous, then the embedding is compact, as the relative compactness in C(K) implies every bounded sequence in X has a subsequence converging in C(K). Equicontinuity of the image of B_X is often quantified via the modulus of equicontinuity \omega(\delta) = \sup \{ |f(x) - f(y)| : d(x,y) \leq \delta, \, f \in B_X \}, which must tend to 0 as \delta \to 0 for the set to be equicontinuous. This modulus controls the uniform control of oscillations, enabling the diagonal argument in the proof of Ascoli–Arzelà to extract convergent subsequences.^[15] A concrete illustration arises in Sobolev spaces: for a bounded domain \Omega \subset \mathbb{R}^n with sufficiently regular boundary (e.g., C^1), the embedding W^{1,p}(\Omega) \hookrightarrow C(\overline{\Omega}) is compact when p > n. Here, Morrey's inequality ensures boundedness and Hölder continuity of functions in the unit ball of W^{1,p}(\Omega), implying equicontinuity, so Ascoli–Arzelà yields the relative compactness of the image. However, the theorem's applicability to embeddings is limited on non-compact domains, such as unbounded sets, where the lack of compactness in the base space K prevents the uniform boundedness and equicontinuity from guaranteeing relative compactness without additional boundary or extension assumptions.

Applications

Sobolev Space Embeddings

In Sobolev spaces W^{k,p}(\Omega), where \Omega \subset \mathbb{R}^n is a bounded domain, the embedding into L^q(\Omega) is compact under specific conditions on the exponents. Specifically, for $1 \leq p < \infty, k \in \mathbb{N}, and kp < n, the embedding W^{k,p}(\Omega) \hookrightarrow L^q(\Omega) is compact when $1 \leq q < p^*, with the Sobolev conjugate exponent defined as p^* = \frac{np}{n - kp}.^[16] This result relies on the domain \Omega possessing sufficient regularity to support extension operators that map functions from \Omega to \mathbb{R}^n while preserving the necessary norms.^[17] The Gagliardo–Nirenberg inequality plays a crucial role in establishing the boundedness of these embeddings, providing estimates of the form \|u\|_{L^{p^*}(\mathbb{R}^n)} \leq C \|\nabla^k u\|_{L^p(\mathbb{R}^n)} for functions with compact support, which extends to bounded domains via localization.^[16] Compactness then follows by combining this boundedness with the properties of extension operators, which allow sequences bounded in W^{k,p}(\Omega) to be handled as if on the whole space, enabling the use of weak convergence arguments to obtain strong convergence in L^q(\Omega).^[17] Compactness holds for domains \Omega that are Lipschitz, as these admit continuous extension operators satisfying the required norm estimates; however, it fails for unbounded domains, such as \mathbb{R}^n, where sequences can escape to infinity via translations without converging strongly in L^q, and for irregular domains like those with fractal boundaries or cusps, where extension operators do not exist or fail to preserve compactness.^[16]^[18] A representative example is the embedding H^1(\Omega) \hookrightarrow L^2(\Omega), which is compact for bounded Lipschitz domains \Omega \subset \mathbb{R}^n with n \geq 1, as sequences bounded in H^1 converge strongly in L^2 due to the control on gradients and the Poincaré inequality on bounded sets.^[16] In the context of variable exponent Sobolev spaces W^{k,p(x)}(\Omega), compact embeddings into variable Lebesgue spaces L^{q(x)}(\Omega) have been established since the early 2000s, requiring the exponent p(x) to satisfy a log-Hölder continuity condition, such as |p(x) - p(y)| \leq \frac{C}{1 + |\log |x - y||} for x, y \in \Omega, to ensure the necessary density and extension properties hold on bounded domains.^[2] This condition prevents rapid oscillations in p(x) that could disrupt compactness, mirroring the role of uniform continuity in constant exponent cases.

Rellich–Kondrachov Theorem

The Rellich–Kondrachov theorem provides a fundamental result on the compactness of embeddings from Sobolev spaces into Lebesgue spaces on bounded domains. Specifically, let \Omega \subset \mathbb{R}^n be a bounded domain with C^1 boundary, k \in \mathbb{N}, and $1 \leq p < \infty. The embedding W^{k,p}(\Omega) \hookrightarrow L^q(\Omega) is compact whenever q < p^*, where the critical exponent is given by

p^* = \frac{np}{n - kp}

for the subcritical case kp < n, or q < \infty for the critical case kp = n. In the supercritical case kp > n, the embedding into L^q(\Omega) is compact for any $1 \leq q < \infty. The embedding is continuous but fails to be compact when q = p^* in the subcritical regime or for q = \infty in the critical regime. This theorem was originally established by Franz Rellich in the 1930s for the Hilbert space setting p=2, using potential-theoretic methods to demonstrate compactness for embeddings of H^k(\Omega) into L^2(\Omega). Rellich's result laid the groundwork for understanding bounded sequences in Sobolev spaces having convergent subsequences in weaker norms. It was extended by Mikhail Mikhailovich Kondrachov in the late 1930s and early 1940s to the general L^p framework, incorporating fractional powers and establishing the precise range of q for compactness via integral comparison techniques. Modern formulations and proofs, including generalizations to various domain classes, appear in Robert A. Adams' 1975 monograph, which consolidates these developments into a comprehensive treatment. A standard proof of the theorem proceeds by first extending functions from W^{k,p}(\Omega) to \tilde{u} \in W^{k,p}(\mathbb{R}^n) using a reflection or cutoff operator that preserves the Sobolev norm up to a constant factor, valid for C^1 domains. Compactness then follows from showing that bounded sets in W^{k,p}(\mathbb{R}^n) are precompact in L^q(\mathbb{R}^n) for the specified q, using appropriate methods to verify the conditions of the Kolmogorov–Riesz criterion, which requires boundedness, uniform decay at infinity, and control of oscillations via translations. Restricting back to \Omega preserves compactness due to the boundedness of the domain.^[19] An important extension of the theorem occurs in the supercritical regime kp > n, where the embedding W^{k,p}(\Omega) \hookrightarrow C^{0,\alpha}(\overline{\Omega}) is compact for $0 < \alpha < k - n/p. This follows from Morrey's inequality, which bounds the Hölder seminorm by the Sobolev norm, combined with Ascoli–Arzelà compactness for equicontinuous families. Such embeddings are crucial for regularity results in elliptic PDEs. The sharpness of the critical exponent p^* = \frac{np}{n - kp} is demonstrated by the failure of compactness at q = p^*, where bounded sequences in W^{k,p}(\Omega) can lack convergent subsequences in L^{p^*}(\Omega). A canonical counterexample involves concentrating functions, such as u_\epsilon(x) = \epsilon^{k - n/p} \phi((x - x_0)/\epsilon) for a fixed smooth \phi \in C_c^\infty(\mathbb{R}^n) with \int |\phi|^{p^*} = 1 and x_0 \in \Omega, scaled so that \|u_\epsilon\|_{W^{k,p}(\Omega)} \leq C as \epsilon \to 0^+, but \|u_\epsilon - u_{\epsilon'}\| _{L^{p^*}(\Omega)} \not\to 0 for distinct small \epsilon, \epsilon' due to disjoint supports or separation. This illustrates the loss of compactness precisely at the threshold.

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