Compactness in Function Spaces

In finite dimensions, boundedness prevents escape to infinity and closedness keeps limit points. In function spaces, a sequence can also escape by oscillating faster, concentrating into narrower regions, or translating its mass away. Compactness criteria identify and block those extra escape modes.

Arzelà–Ascoli for continuous functions #

Let $K$ be a compact metric space and let $\mathcal F\subset C(K)$ with the uniform norm. The family is equicontinuous when

$$\forall\varepsilon>0\ \exists\delta>0\ \forall f\in\mathcal F: d(x,y)<\delta\Rightarrow |f(x)-f(y)|<\varepsilon.$$

The same $\delta$ must work for every function. Arzelà–Ascoli says that $\mathcal F$ is relatively compact in $C(K)$ iff it is equicontinuous and pointwise relatively compact. For real- or complex-valued functions, pointwise relative compactness is equivalent to pointwise boundedness; on compact $K$, equicontinuity plus boundedness at one point already gives uniform boundedness.

Why it works:

1. choose a finite $\delta$-net in $K$;
2. use boundedness to extract a subsequence converging on that finite net;
3. repeat on finer nets and diagonalize;
4. use equicontinuity to extend convergence from net points to all of $K$;
5. completeness of $C(K)$ turns uniform Cauchy behavior into a continuous limit.

Two counterexamples isolate the hypotheses #

The functions $f_n(x)=\sin(nx)$ on $[0,2\pi]$ are uniformly bounded but not equicontinuous: a horizontal displacement of order $1/n$ can change the value by order one. No subsequence converges uniformly.

The functions $g_n(x)=x^n$ on $[0,1]$ are also uniformly bounded but not equicontinuous near $1$. Their pointwise limit is $0$ on $[0,1)$ and $1$ at $1$, which is discontinuous. A uniformly convergent subsequence would have a continuous limit, so none exists.

A reusable sufficient condition #

If every $f\in\mathcal F\subset C^1[a,b]$ satisfies

$$\|f\|_\infty\le M,\qquad \|f'\|_\infty\le L,$$

then the mean-value theorem gives $|f(x)-f(y)|\le L|x-y|$. The family is uniformly bounded and equi-Lipschitz, hence relatively compact in $C[a,b]$. A bounded derivative is therefore a compactness-producing regularity estimate when the target topology forgets the derivative.

Kolmogorov–Riesz for $L^p(\mathbb R^d)$ #

For $1\le p<\infty$, a bounded set $F\subset L^p(\mathbb R^d)$ is relatively compact when two further controls hold uniformly:

Tight tails

$$\lim_{R\to\infty}\sup_{f\in F} \int_{|x|>R}|f(x)|^p\,dx=0,$$

which prevents mass from translating to infinity, and translation continuity

$$\lim_{h\to0}\sup_{f\in F}\|f(\cdot+h)-f\|_p=0,$$

which prevents arbitrarily fine oscillation or concentration. Together with boundedness, these conditions characterize relative compactness in the usual $L^p(\mathbb R^d)$ formulation.

The translated bumps $f_n(x)=\mathbf1_{[n,n+1]}(x)$ have constant $L^p$ norm and identical local shape, but fail tail tightness. Rapid oscillations on a fixed bounded region fail translation continuity. These are distinct escape mechanisms.

Compact embeddings are topology changes #

A set bounded in a stronger norm can be relatively compact in a weaker norm. For example, a uniformly bounded subset of $C^1[a,b]$ is relatively compact in $C[a,b]$, not necessarily in the $C^1$ norm. Compact Sobolev embeddings follow the same pattern: derivative control suppresses oscillation, and the weaker target topology allows subsequences to converge.

When applying a compactness theorem, record:

• the exact domain and target spaces;
• the topology in which boundedness is known;
• the topology in which convergence is claimed;
• the mechanism preventing oscillation, concentration, and translation;
• whether the conclusion is compactness or only relative compactness.