What is equicontinuity? — Mathematics

Definition 7.10 University Mathematics — Year 3 · Chapter 7 — Complete Spaces: Baire, Ascoli, Stone–Weierstrass

A family $\mathcal F \subseteq \mathcal C(K)$ is equicontinuous if for every $\varepsilon > 0$ there is $\delta > 0$ such that

d(x, y) < \delta \implies \norm{f(x) - f(y)} < \varepsilon \quad\text{for \emph{all} } f \in \mathcal F

(one $\delta$ for the whole family — e.g. any family with a common Lipschitz constant, or a common Hölder modulus), and pointwise bounded if $\sup_{f}\norm{f(x)} < \infty$ for each $x$ .

Examples

Example 7.12

The closed unit ball of $\mathcal C(\intcc01)$ is not compact ( $f_n(x) = x^n$ has no uniformly convergent subsequence: the pointwise limit is discontinuous), and indeed $(x^n)$ is not equicontinuous at $1$ . By contrast $\{f : \norm f_\infty \leq 1,\ \operatorname{Lip}(f) \leq 1\}$ is compact: bounded and $1$ -Lipschitz-equicontinuous, and closed. Ascoli explains why compactness fails in infinite dimension (Riesz, Year 2) and what to add to restore it: a uniform modulus of continuity.

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