## § Equicontinuity, Arzela Ascoli

- A sequence/family of functions are said to be equicontinuous if they vary equally in a given nbhd
- Necessary for Arzela Ascoli
- A subset of $C(X)$, space of continuous functions on a compact Hausdorff space $X$ is compact iff if it is closed, bounded, and equicontinuous.
- Corollay: a sequence of $C(X)$ is uniformly convergent iff it is equicontinuous.
- Thus, an equicontinious family converges pointwise, and moreover, since it is uniform convergence, the limit will also be continuous.

#### § Uniform boundedness Principle

The uniform boundedness principle states that a pointwise bounded family of continuous linear operators between Banach spaces is equicontinuous.

#### § Equicontinuity of metric spaces

- Let $X, Y$ be metric space.
- The family $F$ is equicontinuous at a point $p \in X$ if for all $\epsilon > 0$, there is a $\delta > 0$ such that $d(f(p), f(x)) < \epsilon$ for all $f \in F$and all $d(p, x) < \delta$.
- Thus, for a given $\epsilon$, there is a uniform choice of $\delta$ that works for all
*functions *. - It's like uniform continutity, except the uniformity is enforced acrorss the
*functions * $f_i$, not on the points on the domain. - The family $F$ is pointwise equicontinuous iff it is equicontinuous at each point $p$.