Compactly embedded normed spaces

Let $X$ and $Y$ be normed spaces over the same field $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$, read in the real case from A norm on a real vector space, the induced metric, and the dictionary with the metric axioms and in the complex case from Real and complex scalar conventions for normed spaces, and suppose $X\subseteq Y$ with continuous inclusion $J:X\to Y$: there is a real $C\ge 0$ with $\mathrm{\parallel }x{\mathrm{\parallel }}_Y\le C\mathrm{\parallel }x{\mathrm{\parallel }}_X$ for every $x\in X$.

One says that $X$ is compactly embedded in $Y$, written $X⋐Y$, when the inclusion operator $J$ is a compact operator in the sense of Compact linear operator: the image under $J$ of every bounded subset of $X$ (Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space) has compact closure in $Y$ (Open cover, subcover, compact metric space, and compact subset of a metric space).

The sequential form. Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) and the Axiom of Dependent Choice (The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain). Then $X⋐Y$ if and only if every bounded sequence $(x_j)$ in $X$ has a subsequence $(x_{j_k})$ converging in $Y$: Indeed, compact closure gives the sequence conclusion by For a metric space, compact, countably compact, limit point compact, sequentially compact, and complete together with totally bounded are all equivalent, given countable choice and dependent choice. Conversely, fix bounded $B\subseteq X$ and let $K=\overline{J(B)}$. For any sequence $(y_j)$ in $K$, Countable Choice selects $b_j\in B$ with $\mathrm{\parallel }{y}_j-b_j{\mathrm{\parallel }}_Y<1\mathrm{/}j$ (start at $j=1$); a convergent subsequence of $(b_j)$ gives one of $(y_j)$ with the same limit, which belongs to the closed set $K$. Thus $K$ is sequentially compact and For a metric space, compact, countably compact, limit point compact, sequentially compact, and complete together with totally bounded are all equivalent, given countable choice and dependent choice makes it compact. The empty case is immediate. Both readings of $X⋐Y$ are used on this page; the second is the form in which the compactness theorems below are stated.

Continuity of the inclusion is a separate hypothesis and is never inferred from compactness: a compact operator is bounded by Compact linear operator and A bounded linear operator between normed spaces, but the definition above fixes the continuity of $J$ in advance. On this page the continuity of every Sobolev inclusion is verified separately, through the corresponding Sobolev embedding theorem.

• The symbol $X⋐Y$ is used only for pairs of normed spaces embedded in one another as above; it never abbreviates a claim that some particular Sobolev space is compactly embedded in another, which always requires its own theorem with its own domain, exponent and boundary hypotheses.
• If $X$ is finite-dimensional and $Y$ is a normed space containing it with continuous inclusion, then $X⋐Y$: the coordinate isomorphism of A chosen algebraic basis identifies a finite-dimensional normed space with a coordinate space sends a closed bounded coordinate ball to a compact set containing any prescribed bounded subset of $X$, by Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line (identify ${\mathbb{C}}^d$ with ${\mathbb{R}}^{2d}$). Its image in $Y$ is compact by continuity of the inclusion, and the closure of the bounded image is a closed subset of it. The zero-dimensional case is immediate.
• The two formulations agree without any hypothesis on $X$ or $Y$ beyond their being normed spaces, and the choice cost of the passage between them is bounded above by Countable Choice plus Dependent Choice, as recorded in For a metric space, compact, countably compact, limit point compact, sequentially compact, and complete together with totally bounded are all equivalent, given countable choice and dependent choice.