Compactness in Function Spaces

In infinite dimensions, “bounded” does not imply “precompact” (Example 2). To pin down compactness in the two function spaces we care about ( $C(K)$ and $L^p$ ) we need additional structure. Two classical theorems give the right structure: Arzelà-Ascoli for continuous functions, and its $L^p$ analog Kolmogorov-Riesz for integrable functions. Both will be cited repeatedly throughout the course.

Arzelà-Ascoli for $C(K)$ ¶

Theorem 1 (Arzelà-Ascoli)

Let $K$ be a compact metric space and $\mathcal{F} \subset C(K)$ a family of continuous functions. Then $\mathcal{F}$ is precompact (in the sup norm) if and only if

(pointwise bounded) $\sup_{f \in \mathcal{F}} |f(x)| < \infty$ for every $x \in K$ ,
(equicontinuous) for every $\varepsilon > 0$ there exists $\delta > 0$ such that $|f(x) - f(y)| < \varepsilon$ whenever $d(x, y) < \delta$ , for all $f \in \mathcal{F}$ simultaneously.

The two conditions prevent the two ways a sequence of continuous functions can fail to have a uniformly convergent subsequence: the values cannot blow up (boundedness), and they cannot oscillate arbitrarily fast (equicontinuity). A typical use: bounded sequences of $C^1$ functions on a compact interval are equicontinuous because $|f(x) - f(y)| \leq \|f'\|_\infty |x - y|$ , hence precompact in $C^0$ by Arzelà-Ascoli.

Kolmogorov-Riesz for $L^p$ ¶

The same question arises one floor down. Nesting (Remark 2) tells us $L^p(\Omega) \subset L^q(\Omega)$ continuously when $p \geq q$ and $|\Omega| < \infty$ . Is this embedding compact?

The answer is no. On $\Omega = (0, 2\pi)$ , the sequence $u_n(x) = \sin(nx)$ has $\|u_n\|_{L^p} \leq C$ for every $p \in [1, \infty]$ , so it is bounded in every $L^p$ . Yet any two distinct terms satisfy $\|u_n - u_m\|_{L^q} \not\to 0$ : the sequence has no convergent subsequence in $L^q$ , for any $q$ . Integrability alone does not suppress oscillation.

The $L^p$ analog of Arzelà-Ascoli gives a complete answer.

Theorem 2 (Kolmogorov-Riesz-Fréchet)

Let $1 \leq p < \infty$ . A bounded set $K \subset L^p(\mathbb{R}^d)$ is precompact if and only if

(bounded) $\sup_{u \in K} \|u\|_{L^p} < \infty$ ,
(equicontinuous under translation) $\sup_{u \in K} \|\tau_h u - u\|_{L^p} \to 0$ as $h \to 0$ , where $\tau_h u(x) = u(x - h)$ ,
(tight) $\sup_{u \in K} \|u\|_{L^p(|x| > R)} \to 0$ as $R \to \infty$ .

On a bounded domain $\Omega$ , condition (3) is automatic, so precompactness reduces to (1) and (2).

Each condition prevents a specific way mass can escape:

(1) prevents mass from blowing up in size,
(2) prevents mass from escaping into high frequencies, since a function whose translations converge uniformly cannot oscillate arbitrarily fast,
(3) prevents mass from escaping to spatial infinity.

This is the $L^p$ mirror of Arzelà-Ascoli: “bounded + equicontinuous + stays in a compact set” gives precompactness, with $L^p$ -equicontinuity-under- translation playing the role of pointwise equicontinuity.

In later chapters, this theorem is the structural reason Sobolev embeddings produce compactness. A gradient bound of the form $\|\nabla u\|_{L^p} \leq M$ supplies condition (2) automatically: by the fundamental theorem of calculus, $\|\tau_h u - u\|_{L^p} \leq |h|\,\|\nabla u\|_{L^p}$ . So the Rellich-Kondrachov theorem is Kolmogorov-Riesz applied to the Sobolev ball, with the equicontinuity condition upgraded from a hypothesis to a consequence of gradient control.

Arzelà-Ascoli for C(K)C(K)C(K)¶

Theorem 1 (Arzelà-Ascoli)

Kolmogorov-Riesz for LpL^pLp¶

Theorem 2 (Kolmogorov-Riesz-Fréchet)

Arzelà-Ascoli for $C(K)$ ¶

Kolmogorov-Riesz for $L^p$ ¶